What is the difference:
-I want to drink something.
-Me too.
And
-I want to drink something.
-Me either.
What is the difference:
-I want to drink something.
-Me too.
And
-I want to drink something.
-Me either.
The first conversation is grammatically correct. It means that person A wants to drink something. Person B agrees and also wants to drink something.
Your second conversation is incorrect though. 'Me either' is not a valid form in English. The following would be correct if you didn't want to drink something:
-I don't want to drink something
-Me neither