Can I run both of these mods at once? If not, which is "better"?

All Natural has AWS integrated into it. You'll have to choose one or the other. This goes for most weather mods (i.e. Natural Weather, Enhanced Weather, etc.).

If you install the All Natural OMOD version with Oblivion Mod Manager, then you will choose which weathers you want - the choices are Natural Weather, Atmospheric Weather System and Enhanced Weather. You can choose just one, any combination of two, or all three. Therefore, if you got All Natural - you don't need AWS at all.

Didn't I already answer this? In any case, the above are correct. Also, a little reading of the mod's description and readme, both of which are viewable at TES Nexus without needing to download anything, would have answered that question without the need for a thread.