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.
Common usage in English is to say "I'm finished" instead of the correct "I have finished"... even if having finished whatever you were doing makes you feel as though you are finished. =) Being finished implies you are dead. Having finished means you have completed your task.
I don't understand the pronunciation of "we finished"? On the audio, "we finished" sounds like siyamenu, but on Pealim.com it says "we finished" is siyamnu. https://www.pealim.com/search/?q=%D7%A1%D7%99%D7%99%D7%9E%D7%A0%D7%95 Which pronunciation is correct for "we finished"?