“Je ne bois pas de vin” is a generic statement : I don’t drink wine, I don’t like the taste. It’s often said alone, you don’t really need to add what you drink instead (and you probably don’t always drink apple juice every time other people drink wine.

“Je ne bois pas du vin” refers to what you have in your glass right now : I’m not drinking wine, it’s just grape juice. It makes more sense if it’s followed by the correct information.

It doesn’t make sense to say you’re drinking apple juice when you say you don’t drink wine as a general statement, so it implies you’re talking about what you’re drinking right now. It’s textbook logic, you have to deduce context from the little information you’re given. It’s easier in a normal conversation, don’t worry.

The second sentence still sounds weird and a little academic for a native. It’s a little too formal, especially in a context where you could be drinking wine.

The sentences mean different things.

Je ne bois pas du vin, mais du jus de pomme.

translates best to English as

What I am drinking isn’t wine, it’s apple juice.

On the other hand,

Je ne bois pas de vin, mais du jus de pomme.

translates best to English as

I don’t drink wine, but [I do drink] apple juice.

The negated partitive article emphasizes that the identity of something is not wine. For the same reason, the verb être always takes the full negative partitive and not just a “de”:

Ceci n’est pas du vin.

Finally, I don’t think I would word the second sentence this way in French; I’d prefer to repeat the subject somehow:

Je ne bois pas de vin, mais parfois je bois du jus de pomme.

However, I am not a native French speaker and this last intuition may be wrong.