Is there an age difference between the users of לגמור and לסיים? That is, do older speakers tend to use לגמור where younger speakers might increasingly prefer לסיים?
i feel that religious people would use סיים as reference to biblical application, like the end of something or finishing a talmud. לגמור is a more modern Hebrew way of saying done.
I've always gotten לגמור and לסיים mixed up. When does one use one over the other?
They mean the same but not used exactly the same - formally, you don't use לסיים with an infinitive: גמרתי את הספר and סיימתי את הספר are both correct. גמרתי לקרוא is correct, סיימתי לקרוא not so much. But I grudgingly admit that it is a very common mistake and only kibizers like me mind it.
One other thing, לסיים is also "graduate", so if the thing you "finish" is a course, a school, a university, then סיימתי is very appropriate.
There isn't really a difference, and they can be used interchangeably. When using "finish + noun" (e.g. to finish the food) the word לגמור used to be preferred, but this is changing.
The word לגמור also has a sexual meaning in slang (same as "to come" in English), so in recent years people tend to use לסיים in more and more cases where לגמור would previously be used.