Asymmetric vs Symmetric Encryption
Symmetric and asymmetric encryption are distinct operations which operate in distinct contexts, for distinct usages and distinct attack models. There is little sense in claiming that either is "more secure" than the other (it would first require some kind of quantified measure of security, which is not easy to define).
Asymmetric encryption is more demanding: it is about realizing encryption and being able to publish the ways to encrypt (the public key) without revealing the ways to decrypt (the private key). A practical implementation needs mathematics, whereas symmetric encryption is mostly scrambling things around (mind you, it is difficult to do a good job of scrambling; but still, less mathematics are involved).
Commonly deployed systems (e.g. SSL/TLS) combine asymmetric encryption and symmetric encryption (and a few other things as well) into protocols which do some intended job (e.g. "bidirectional tunnel with confidentiality, integrity and authentication").
There is a sense in which you can define the strength of a particular encryption algorithm¹: roughly speaking, the strength is the number of attempts that need to be made in order to break the encryption. More precisely, the strength is the amount of computation that needs to be done to find the secret. Ideally, the strength of an algorithm is the number of brute-force attempts that need to be made (weighed by the complexity of each attempt, or reduced if some kind of parallelization allows for multiple attempts to share some of the work); as attacks on the algorithm improve, the actual strength goes down.
It's important to realize that “particular encryption algorithm” includes considering a specific key size. That is, you're not pitching RSA against AES, but 1024-bit RSA (with a specific padding mode) with AES-256 (with a specific chaining mode, IV, etc.). In that sense, you can ask: if I have a copy of my data encrypted with algorithm A with given values of parameters P and Q (in particular the key size), and a copy encrypted with algorithm B with parameters P and R, then which of (A,Pval₁,Qval₁) and (B,Pval₂,Rval₂) is likely to be cracked first?
In practice, many protocols involve the use of multiple cryptographic primitives. Different primitives have different possible uses, and even when several primitives can serve a given function, there can be one that's better suited than others. When choosing a cryptographic primitive for a given purpose, the decision process goes somewhat like this:
- What algorithms can do the job? → I can use A or B or C.
- What strength to I need? → I want 2N operations, so I need key size LA for primitive A, LB for primitive B, LC for primitive C.
- Given my constraints (brute speed, latency, memory efficiency, …), which of these (LA-bit A or LB-bit B or LC-bit C) is best?
For example, let's say your requirement is a protocol for exchanging data with a party you don't trust. Then symmetric cryptography cannot do the job on its own: you need some way to share the key. Asymmetric cryptography such as RSA can do the job, if you let the parties exchange public keys in advance. (This is not the only possibility but I won't go into details here.) So you can decide on whatever RSA key length has the right strength for your application. However RSA is slow and cumbersome (for example there aren't standard protocols to apply RSA encryption to a stream — mainly because no one has bothered because they'd be so slow). Many common protocols involving public-key cryptography use it only to exchange a limited-duration secret: a session key for some symmetric cryptography algorithm. This is known as hybrid encryption. Again, you choose the length of the session key according to the desired strength. In this scenario, the two primitives involved tend to have the same strength.
¹ The same notion applies to other uses of cryptography, such as signing or hashing.
Choosing between symmetric and asymmetric encryption depends on the use case.
Symmetric encryption is used when a *small group of users need access to the information. Furthermore symmetric encryption is nice because it is easier to understand (less likely to mess it up) and the algorithms tend to be faster.
Asymmetric encryption is used when a large group of users need access to the information. Furthermore, asymmetric cryptography can be used in reverse to sign documents. This is especially interesting because it allows people to certify that a public key belongs to a certain person.
A group of 10 users requires 45 unique keys (9+8+7+6+5+4+3+2+1), to allow every pair of users to communicate securely. Now consider the Internet instead of the small group of 10 people. It's obvious that this cannot be handled with symmetric keys.
In the real world, both types are often combined. An asymmetric approach is used to confirm the identity of a communication partner and to transmit something that will result in a symmetric key. This symmetric key is then used for performant encryption of the actual data.