Learn Croatian: Ne čujem ni grmljavinu ni vjetar. - I hear neither thunder nor the wind.

Croatian requires double negation here. When you use the conjunction pair ni... ni... (neither... nor...), the main verb must also be negative. So you literally say 'I do not hear neither thunder nor wind.' If you drop the Ne, the sentence is grammatically incorrect.

Both thunder and wind are the objects being heard, so why does grmljavina change to grmljavinu while vjetar doesn't change?

Both nouns are actually in the accusative case because they are the direct objects of the verb čujem (I hear). Grmljavina is a feminine noun, so its ending changes from -a to -u. Vjetar is a masculine inanimate noun (not a living being), and the accusative form for those is identical to their dictionary (nominative) form.

Could I just say Ne čujem grmljavinu i vjetar instead?

You could, but it changes the meaning slightly. Ne čujem grmljavinu i vjetar simply means 'I do not hear the thunder and the wind.' Using ni... ni... makes it more emphatic, specifically stressing that you hear neither of the two distinct sounds.