Both can be fine. While the first focuses more on the objective description of the weather, and the second focuses more on someone's subjective opinion of the weather, the answer can go either way, depending on how the listener chooses to interpret the question.

Examples:

James: What's the weather out there?
Phil: It's miserable.
James: No, I mean what's it like? Warm, sunny, rainy ...?
Phil: It's hot, humid, and totally miserable.

James: How's the weather out there?
Phil: Sunny, some clouds, relatively cool.
James: So it's nice?
Phil: Yes, it's nice.

For this reason, I wouldn't worry too much about it. You can always clarify your question if you want to know something specific.

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.

I agree with both the other answers about the relative usage of the two forms you've mentioned. I will add one more possibility:

Often when I'm deciding what to wear for the day I'll ask my spouse to look at his phone and tell me what his weather app says. In that case, I'll usually use some variation on

What is the weather supposed to be today?
What is the weather going to be (like) today?

or just

What's the (weather) forecast for today?

In this case, I'm not talking about the weather right now but rather the (expected) conditions throughout the day.

To me "What is the weather today?" is a very unnatural sentence. At least in my recent memory, I've never heard a native English speaker say that.

As others have mentioned, the what version conveys a more scientific tone, while the how version conveys a subjective tone. As a result, I think you would be more likely to use what to discuss forecasts and how to discuss the current weather. However, you will also hear "What's the weather like today?" and I would say this like/what pairing is semantically equivalent to how.

I'm an American English speaker (from California), so I would be curious to see if other people who use the phrasing "What's the weather today?" come from a different area, because it sounds so unnatural to me (though of course it's grammatically correct).

As an English man I have just come across this phrase in a foreign school teaching children English. I have to admit I have never heard anyone say or use the phrase, "What is the weather today ?". In England we would say, "What is the weather like today?" or " What is the weather forecast for today?". I hope that helps.