Which is which? Am I right in thinking that "talar" is typically closer to "talk" and "pratar" to "speak"?
Other than set phrases ("talking shop", "public speaking", "talk the talk", "speak a foreign language", etc.), I can't think of any differences in English.
Do you have any Swedish examples of where one would sound natural and the other wouldn't?
The other way around, actually. Tala = speak, prata = talk.
When it comes to speaking a language, it's often more common and fitting to use "tala". When delivering a speech, you also use "tala". For common chatting and talking, use "prata". Just like speak, "tala" sounds a little bit more serious.