"By all means" is more gracious. It's like saying "certainly, absolutely, most definitely you may". "Yes, you may" simply grants permission. You can make it more gracious by adding an embellishment, such as "Yes, please, certainly you may".

Yes it is stronger, more emphatic. "Can I make a suggestion?" Yes, of course! By all means!

Great question. And, in short, yes...kind of "I need you to complete the project by Friday, at all costs / ny all means NECESSARY" Note the word 'necessary'. "At all costs" obviously means...no matter what money you need/no matter what the consequences. "By all means necessary" means...whatever methods you need.

Exactly what Jonathan has explained, both "by all means" and "you may" grant permission however "by all means" is a lot more enthusiastic and encouraging.