We use both. However, asking, "What's the weather like?" is asking about the conditions right now. Asking, "What's the weather look like?" refers more to future weather conditions. "Does it look like it's going to rain? Will it stay sunny? Do you think it will stop raining?" These things and any other weather condition that is possible are all implied by asking, "What's the weather look like?"

I've heard and said both of these. You can say "What's the weather like?" to ask about the weather that day, in the future, or in general. For example, if you are planning a vacation, you might ask the travel agent "What's the weather like?". "What does the weather look like?" is used to ask someone about the weather when you are planning something more immediate. Maybe the person can actually see what it looks like outside or knows the forecast. For example, if you're planning a picnic with someone, you might ask "What does the weather look like tomorrow?" Or if you are talking with a friend across town, you might say "What's the weather look like over there?" I think that this one is not used to ask about general climate. You wouldn't say "What does the weather look like in Hawaii?", but you could say "What's the weather like in Hawaii?".