Why is mist gray but water clear?

Mist is a suspension of tiny water droplets in air.

Light traveling through the mist gets randomly scattered, mainly by bouncing of the droplets. That makes mist far less transparent than bulk water.

I don't think mist is literally gray in colour but the fact that mist is far less transparent than pure air (or bulk water) causes it to look the way it does.

Other suspensions like smoke (a suspension of tiny solid particles in air) look quite similar, due also to light scattering. Another example is very much diluted milk (an emulsion of fat droplets in water, mainly).

Regarding water droplets collecting on the surfaces of your glasses:

Those water droplets backscatter the incoming light in random directions, including ones away from your eyes. This means that any glass lens surface populated with water droplets will appear less bright than it would without the droplets, and the random scattering will obliterate anything you might otherwise be able to discern through the lens. In addition, small droplets crowded closely together cannot be resolved by your eye as individual droplets. Result: uniform smooth gray appearance.

As the water droplets begin to merge, you start being able to see objects through the lens and through the water droplets and the result is less dim (gray).

There are two factors at work here. The first is whether water is "clear", meaning, allows most light to pass through it. And it is, whether it's solid or in a bunch of tiny droplets.

The second is whether you can see through it. In order for a substance to be "transparent", it has to have two properties. First, it has to allow a non-trivial percentage of light to pass through. Materials may absorb a significant percentage of light and still be transparent (e.g. apple juice). In order to see through a material, however, there must be limited scattering; that is, the photons passing through must retain the same relative orientation, otherwise the image you see will be distorted, like a picture taken with an out-of-focus lens, only worse.

This phenomena is actually all around you. To start with, consider a lake. You probably know that if the lake is very still, you can often see through the water. As waves form, what you can see becomes more and more distorted. Now, if you imagine those waves as being really tiny, you start to understand what's happening when you look at mist. It's not that light isn't passing through the water droplets, it's that it's being scattered (refracted) in all directions such that you can't see a clear image. Again, like an out-of-focus photograph, only more so.

You don't just see this in water, either. The difference between "regular" and "frosted" glass is exactly the same; a smooth or (microscopically) rough surface either allows a clear image or a very blurry one. You also see something similar in metals; a rough surface is "shiny" but doesn't produce a clear reflection, but with enough polishing — that is, increasing the surface smoothness — you can get a "mirror finish".

As to the second part of your question, water, due to surface tension, naturally has a very smooth surface. As water collects and transitions from many small droplets with lots of scattering, you wind up with a smoother surface and less scattering.

As to why it's gray... that's the simplest of all. "Gray" is a uniform mixture of all colors of light. By definition (given the way our eyes naturally "white balance" for ambient light), the average of all light you are seeing at any time is gray. If you take all the light in a given environment and mix all those photons together, you will always get gray. Similarly, if you take just about any photograph and blur the ever loving snot out of it, you'll get gray. (This works because water, at least in the sorts of quantities we're talking about here, does not significantly absorb any particular wavelength more than others. If your mist instead absorbs most blue light, you'd get yellow, and so forth.)