They're both perfectly natural. Arguably some people might think the what version is more appropriate when the speaker is specifically interested in knowing what the weather actually is (or perhaps will be, later in the day).
Conversely, the how version might be more likely if what the speaker wants to know is how the addressee feels about the weather.
Expanding on the above, I suspect the "frequency of occurrence" of the how version (relative to the what version) would be higher in the context of telephone calls. If you're talking to someone who's far enough away that "their" weather is likely to be different to whatever you're currently experiencing, you'd have more reason to ask what they think of their weather.
But if you're talking to someone who's actually with you, you probably wouldn't be asking what the current state of the weather is (you can see as well as them whether it's raining or not). And if you're asking What is the weather [forecast for] today?, that would rarely be phrased using how.