I'm at the proverbial izakaya and my friend tells me one is more polite. I think theyre the same. Who's right? When can I use one or the other?
すみません is the correct pronunciation, so on that count I suppose it would be considered more polite by a very small margin. But just like in English, people can be lazy with pronunciation, which is why you'll sometimes hear すいません. You also might run across すんません and すんまへん, but from what I've read these are more Kansai-isms than anything else. When in doubt, stick with the standard pronunciation, but in all cases the difference is in pronunciation only (the meaning stays the same).
If you compare these two phrases, すみません is more formal than すいません, and therefore probably more polite.
すみません is literally the polite form of the negation of すむ (済む). 済む means to be finished or completed.
宿題が済んだら遊びに行く。 (しゅくだいがすんだらあそびにいく。) Once the homework is done, I will go out to play.
すみません, whose literal meaning would be “It will not be finished,” means apology probably because the speaker admits that what he/she did requires some follow-up action.
すいません (suimasen) is the result of dropping the consonant “m” in すみません (sumimasen) and is less formal than すみません.