"What" is about (near) infinite amount of possibilities.
What indicates that spring is approaching? Warm weather, chicks, flowers popping up, my calendar flipping to March, people wearing short pants, days being longer. Many things.
But if you say "the weather is warm" that shrinks the possibility space from infinite to just that one thing. Not that that one thing exclusively indicates spring, but it's the only thing in the sentence, and thus the only thing in the sentence that the second part of the sentence can point back at.
Use "which" instead.
"Which" is used for a smaller amount and absolutely certain things.
Like I already mentioned, the second part of the sentence points back to the first part, so you've only got 1 thing to 'choose' from.
Your sentence, improved.
There are a few other errors in your sentence. I'd rewrite it as one of the following:
The weather is getting warm, which indicates that Spring is approaching.
The weather is getting warm, which is indicative of Spring approaching.
I'd personally go with the first, as it's more direct.