What is the difference between sumimasen and gomennasai?
Feel free to just provide example sentences.

"Sumimasen (すみません)" sounds more formal than "gomennasai (ごめんなさい)."

You may also hear "suimasen (すいません)." Suimasen (すいません) is less formal than sumimasen (すみません) and quite often used in everyday situations.

Please note that sumimasen (or suimasen) is also used to mean "excuse me." For example, you can say sumimasen (or suimasen) when you want to ask someone a question, say the way to the station, on the street.

Personally, when I want to draw someone's attention, say when I want to catch a store clerk, I would more often use suimasen (すいません) than sumimasen (すみません) because suimasen (すいません) is just easier to say. I feel that a lot of people use suimasen (すいません) to mean "excuse me" in everyday situations.

When I want to apologize formally, I would say, sumimasen (すみません) rather than suimasen (すいません) because sumimasen (すみません) sounds more formal. If I want to apologize more formally, I would say, "moushiwake arimasen (もうしわけありません)." Some may say, "moushiwake gozaimasen (もうしわけございません) to mean the same, which may sound more formal than moushiwake arimasen (もうしわけありません). Oh, and you may also hear "moushiwake nai (もうしわけない)" or "moushiwake nai desu (もうしわけないです)." Moushikake nai desu (もうしわけないです) sounds more polite than just moushiwake nai (もうしわけない). And moushiwake nai desu (もうしわけないです) sounds less formal than もうしわけありません and more formal than sumimasen (すみません). In addition, you can make those expressions more formal by adding adverbs like makotoni (まことに), taihen (たいへん), or hontouni (ほんとうに) at the beginning of the phrase. So, there are many variations, but I think the most basic ones can be listed as follows (formal to informal order):

Moushiwake gozaimaen. (もうしわけございません)
Moushiwake arimasen. (もうしわけありません)
Moushiwake naidesu. (もうしわけないです)
Moushiwake nai. (もうしわけない)
Sumimasen. (すみません)
Suimasen. (すいません)
Gomennasai. (ごめんなさい)
Gomen. (ごめん)

I hope I'm not confusing you...