Why are protons heavier than electrons?
There are multiple reasons why protons are heavier than electrons. As you suggested, there are empirical and theoretical evidence behind this. I'll begin with the empirical, since they have important historical context associated with them. As a preface, this will be a fairly long post as I'll be explaining the context behind the experiments and the theories.
Empirical Electron Mass
Measuring the mass of an electron historically is a multi-step process. First, the charge is measured with the Millikan oil drop experiment, then the charge-to-mass ratio is measured with a variation of J.J. Thomson's experiment.
Millikan Oil Drop
In 1909, Robert A. Millikan and Harvey Fletcher measured the mass of an electron by suspending charged droplets of oil in an electric field. By suspending the oil droplets such that the electric field cancelled out the gravitational force, the charge of the oil droplet may be determined. Repeat the experiment many times for smaller and smaller oil droplets, and it may be determined that the charges measured are integer multiples of a singular value: the charge of an electron.
$$e = 1.60217662 \times 10^{-19} \, \mathrm{C}$$
J.J. Thomson's Experiments
In 1897 J. J. Thomson proved that cathode rays (a beam of electrons) were composed of negatively charged particles with a massive charge-to-mass ratio (as compared to ionized elements). The experiment began with first determining if cathode rays could be deflected by an electric field. The cathode ray was shot into a vacuumed Crookes tube, within which it'd pass between two plates before impacting an electric screen. When the plates were charged, the beam would deflect and hit the electric screen thereby proving that cathode rays contained a charge.
Later he would perform a similar experiment, but exchange the electric field for a magnetic field. This time though, the magnetic field would induce centripetal acceleration upon the cathode ray and produce circles. By measuring the radius of the circle and the strength of the magnetic field produced, the charge-to-mass ratio ($e/m_e$) of the cathode ray would be obtained.
$$e/m_e = 1.7588196 \times 10^{11} \, \mathrm{C} \cdot \mathrm{kg}^{-1}$$
Multiply this by the elementary charge obtained in the Millikan oil experiment, and account for uncertainty, and the mass of the electrons in the cathode ray is obtained.
$$m_e = \frac{e}{\frac{e}{m_e}} = \frac{1.60217662 \times 10^{-19} \, \mathrm{C}}{1.7588196 \times 10^{11} \, \frac{\mathrm{C}}{\mathrm{kg}}} = 9.10938575 \times 10^{-31} \, \mathrm{kg}$$
Empirical Proton Mass
Ernest Rutherford is credited with the discovery of the proton in 1917 (reported 1919). In that experiment he detected the presence of the hydrogen nucleus in other nuclei. Later he named that hydrogen nucleus the proton, believing it to be the fundamental building block to other elements. Since ionized hydrogen consisted only of a proton, he correctly deduced that protons are fundamental building blocks to the nuclei of elements; however, until the discovery of the neutron, ionized hydrogen and the proton would remain interchangeable. How then, was the proton mass measured? By measuring the mass of ionized hydrogen.
$$m_p = 1.6726219 \times 10^{-27} \mathrm{kg}$$
This is done in one of several ways, only one of which I'll cite here.
J.J. Thomson Variation
Repeat J.J. Thomson's experiment with magnetic deflection; but, swap out the cathode ray for ionized hydrogen. Then you may measure the charge to mass ratio ($e/m$) of the ions. Since the charge of a proton is equivalent to the charge of an electron:
$$m_p = \frac{e}{\frac{e}{m}} = \frac{1.60217662 \times 10^{-19} \, \mathrm{C}}{9.5788332 \times 10^{7} \, \frac{\mathrm{C}}{\mathrm{kg}}} = 1.67262 \times 10^{-27} \, \mathrm{kg}$$
Other variations
Other variations may include the various methods used in nuclear chemistry to measure hydrogen or the nucleus. Since I'm not familiar with these experiments, I'm omitting them.
Empirical Proton to Electron Mass Ratio
So now we've determined: $$m_p = 1.6726219 \times 10^{-27} \, \mathrm{kg}$$ and $$m_e = 9.10938575 \times 10^{-31} \, \mathrm{kg}$$
Using the two values and arithmetic:
$\frac{m_p}{m_e} = \frac{1.6726219 \times 10^{-27} \, \mathrm{kg}}{9.10938575 \times 10^{-31} \, \mathrm{kg}} = 1836$, or $1800$ if you round down.
Theoretical Proton to Electron Mass Ratio
Theoretically, you first need to understand a basic principal of particle physics. Mass and Energy take on very similar meanings in particle physics. In order to simplify calculations and use a common set of units in particle physics variations of $\mathrm{eV}$ are used. Historically this was developed from the usage of particle accelerators in which the energy of a charged particle was $\mathrm{qV}$. For electrons or groups of electrons, $\mathrm{eV}$ was convenient to use. As this extends into particle physics as a field, the convenience remains, because anything develop theoretically needs to produce experimental values. Using variations of $\mathrm{eV}$ thus removes the need for complex conversions. These "fundamental" units, called the planck units, are:
$$\begin{array}{|c|c|c|} \hline \text{Measurement} & \text{Unit} & \text{SI value of unit}\\ \hline \text{Energy} & \mathrm{eV} & 1.602176565(35) \times 10^{−19} \, \mathrm{J}\\ \hline \text{Mass} & \mathrm{eV}/c^2 & 1.782662 \times 10^{−36} \, \mathrm{kg}\\ \hline \text{Momentum} & \mathrm{eV}/c & 5.344286 \times 10^{−28} \, \mathrm{kg \cdot m/s}\\ \hline \text{Temperature} & \mathrm{eV}/k_B & 1.1604505(20) \times 10^4 \, \mathrm{K}\\ \hline \text{Time} & ħ/\mathrm{eV} & 6.582119 \times 10^{−16} \, \mathrm{s}\\ \hline \text{Distance} & ħc/\mathrm{eV} & 1.97327 \times 10^{−7} \, \mathrm{m}\\ \hline \end{array}$$
Now then, what's the rest energies of a proton and electron?
$$\text{electron} = 0.511 \, \frac{\mathrm{MeV}}{c^2}$$
$$\text{proton} = 938.272 \, \frac{\mathrm{MeV}}{c^2}$$
As we did with the experimentally determined masses,
$$\frac{m_p}{m_e} = \frac{938.272 \, \frac{\mathrm{MeV}}{c^2}}{0.511 \, \frac{\mathrm{MeV}}{c^2}} = 1836$$
which matches the previously determined value.
Why?
I'll preface this section by pointing out that "why" is a contentious question to ask in any science without being much more specific. In this case, you may be wondering what causes the proton mass to 1800× larger than the electron. I'll attempt an answer here:
Electrons are elementary particles. They cannot (or at least have never been observed to) break down into "constituent" particles. Protons, on the other hand, are composite particles composed of 2 up quarks, 1 down quark, and virtual gluons. Quarks and gluons in turn are also elementary particles. Here are their respective energies:
$$\text{up quark} = 2.4 \, \frac{\mathrm{MeV}}{c^2}$$
$$\text{down quark} = 4.8 \, \frac{\mathrm{MeV}}{c^2}$$
$$\text{gluon} = 0 \, \frac{\mathrm{MeV}}{c^2}$$
If you feel that something is off, you're correct. If you assume
$$m_p = 2m_{\uparrow q} + m_{\downarrow q}$$
you'll find:
$$m_p = 2m_{\uparrow q} + m_{\downarrow q} = 2 \times 2.4 \, \frac{\mathrm{MeV}}{c^2} + 4.8 \, \frac{\mathrm{MeV}}{c^2} = 9.6 \, \frac{\mathrm{MeV}}{c^2}$$
but
$$9.6 \, \frac{\mathrm{MeV}}{c^2} \ne 938.272 \, \frac{\mathrm{MeV}}{c^2}$$
This begs the question: what happened, why is the proton mass 100 times larger than the mass of its constituent elementary particles? Well, the answer lies in quantum chromodynamics, the 'currently' governing theory of the nuclear force. Specifically, this calculation performed above omitted a very important detail: the gluon particle field surrounding the quark that binds the proton together. If you're familiar with the theory of the atom, a similar analogy may be used here. Like atoms, protons are composite particles. Like atoms, those particles need to be held together by a "force".
For atoms, the Electromagnetic Force binds electrons to the atomic nucleus with photons (who mediate the EM force). For protons, the Strong Nuclear Force binds quarks together with gluons (who in turn mediate the SN force). The difference between the two though, is that photons can exist independently of the electron and nucleus. Thus we can detect it and perform a host of measurements with them. For gluons though, they not only mediate the strong force between quarks, but may also interact with each other via the Strong Nuclear Force. As a result, strong nuclear interactions are much more complex than electromagnetic interactions.
Gluon Color Confinement
This goes further. Gluons carry a property called color. When two quarks share a pair of gluons, the gluon interaction is color constrained. This means that as the quarks are brought apart, the 'color field' between them increases in strength linearly. As a result, they require an ever increasing amount of energy to be pulled apart from each other. Compare this to the EM force. When you try to pull an electron from its atom, it requires enough energy to be plucked from its shell into the vacuum. If you don't, it'll jump up one or more energy levels, then fall back to its original shell and release a photon that carries the difference.
Similarly, if you want to pluck an object from a planet, you need to provide it with enough energy to escape the planet's gravity indefinitely (energy needed to reach escape velocity). Unlike the gravitational force and the electromagnetic force, the force binding gluons to each other grows stronger as they grow apart. As a result, there comes an inevitable point where the it becomes increasingly more energetically favorable for a quark-antiquark pair to be produced than for the gluons to be pulled further. When this occurs, the quark and antiquark bind to the 2 quarks that were being pulled apart, and the gluons that were binding them are now binding the new pair of quarks.
This animation is from Wikipedia, courtesy of user Manishearth under the Creative Commons Attribution-Share Alike 3.0 Unported license.
But wait! Where did those two quarks came from? Recall how pulling the quarks apart requires energy? Well that energy is on the scale of $\mathrm{GeV}$. At these scales, the energy may convert to particles with kinetic energy. In fact in particle accelerators, we typically see jets of color neutral particles (mesons and baryons) clustered together instead of individual quarks. This process is called hadronization but is also referred to as fragmentation or string breaking depending on the context or year. Finally I must point out that this one of the least understood processes in particle physics because we cannot study or observe gluons alone.
Proton Mass
So, now going back to the original question. Earlier we noticed that the empirical proton mass was $938.272 \, \frac{\mathrm{MeV}}{c^2}$; but, theoretically its mass should be $9.6 \, \frac{\mathrm{MeV}}{c^2}$. The $928.672 \, \frac{\mathrm{MeV}}{c^2}$ difference arises from the color constraints that binds the three quarks together. In simpler terms: the nuclear binding energy of the proton.
As noted "why" is a tricky question but we may ask what is the most fundamental view known concerning this question.
Electrons and protons are very different beasts. Electrons as far as we can tell are elementary, participating in the electromagnetic and so-called weak interactions. On the other hand protons are known to consist of quarks. Quarks are very similar in many properties to electrons but unlike the latter they also participate in the so-called strong interaction described by the theory called quantum chromodynamics (QCD).
For reasons I won't elucidate here the strong interaction works like a rubber band between quarks permitting them to behave as if they were free on very short distances (which we can see on collider experiments from which we know about their existence) but growing stronger and stronger with distance, so quarks never fly around as free particles, only in the form of the composite particles known as hadrons: protons, neutrons, pions etc.
In addition to the quark masses (which are quite small actually) the proton gets its mass from their interaction energy. Because the strong interaction is (surprise) very strong, this energy is huge, constituting almost 99% of its mass. Now can we calculate it using QCD? This is an extremely hard problem - QCD is easy in the regime when quarks are almost free and the strong interaction can be treated as a perturbation. But to compute protons' mass we need to work in a completely different regime for which most computational methods are useless. However it was successfully done using lattice QCD with an error of less than 2%.
It's just an empirical value. According to our current knowledge, the masses actually come from some more fundamental quantities - the electron yukawa coupling and the Higgs field vev, in the case of the electron mass; and the QCD confinement scale (which in turn comes from the strong coupling constant), in the case of the proton mass. But where those numbers come from, we don't know.