"All I need" is used when you want to express the fact that what you need isn't much. You might say:
- "All I need is one more minute!"
- "All you need is love!"
"What I need", on the other hand, is used when you're simply trying to explain what you need, not how important the thing(s) are. You could have a list, or be in a position where what you need is irrelevant, all that matters is that you get something done, for example.
- "What I need is some more time."
- "I need his prosthetic leg."
- "I'll need several million pounds."
So, in summary, use "all" when you want to emphasize that you don't need much, and "what" when you just want to tell someone what you want.
As @WhatRoughBeast said, beware of this being said ironically, or sarcastically. I'm not sure if this is difficult to hear in someone's voice when coming from a different language, though!