"I don't want beer."
Translation:Nem kérek sört.
Yes, that has a slightly different meaning, it is more polite but meanwhile it suggests uncertainity, too. (It depends on the context whether the polite or the uncertain overtone is stronger.) "Nem szeretnék sört" would be "I wouldn't like beer", and it also implies that you would like to have something else, like "nem szeretnék sört, de kávét igen!" (I wouldn't like beer but coffee). For uncertainity part, it is better to explain a dialogue:
"Come on, let's have a beer!"
"I wouldn't like beer."
"Okay, what would you like instead?"
"Dunno. An ice cream?"
– Gyere, igyunk egy sört!
Note this is not word by word but idiomatic translation!
– Nem szeretnék sört.
– Oké, mit szeretnél inkább?
Lit. Oké, mit szeretnél helyette? but this is rare in spoken language
– Nem'tom. Fagyit...
Nem'tom = Nem tudom: rather informal. Note that that type of hesitation that is often written in English with question mark at the end of a statement, we express with ellipsis. Also dialogues are written with n-dash ("gondolatjel") in front, without quotation marks. In most cases n-dash is followed by a space, but this sometimes omitted.