From this answer I can see that they're both casual, but ごめん is more casual.
There's also this question but it's rather about な and ね generally. Still it (or this article) explains the uses of ね, and my guess is, ごめんね is a more friendly or emotional way to say "sorry"? And ごめん is more calm, and maybe abrupt? Does that sound correct?
A very simplified explanation is that this type of ね adds emotion and sincerity. So you can think of ごめん as closer to "(I'm) sorry" and ごめんね as closer to "I'm so sorry". To your friend, you could say ごめん for a trivial (or "casual") mistake, but you probably want to say ごめんね, 本当にごめんね, and so on for a relatively serious mistake like pouring water on their clothes.
However, note that ね is just one of many sentence-ending particles, and not everyone uses it. ね tends to sound feminine, childlike and/or gentle. While ね is often the safest option for Japanese language learners, those who use more masculine language (such as those preferring 俺 over 僕/私) might also prefer expressions like ごめんな, 悪い, すまん, and so on. It is almost impossible for someone like Goku from Dragon Ball to say ごめんね.
(All of the expressions listed above are casual/informal.)
The particle ね is used at the end of sentences to mean that the speaker has some doubt about whatever it is they say. However, more often then not it is translated as "...., right?" in English which for the most part hits the mark because often the doubt is there to have the listener verify what you said.
With that in mind, as far as I know ね adds here a request for accepting your apology. A literal translation to English would be somewhat unnatural: "Sorry, ok?" - It almost sounds aggressive but in Japanese I think this does the opposite, it makes it more polite and I would translate ごめんね as "Pardon" or "Excuse me" or even "Please excuse me" while ごめん would be something like just "Sorry" while barreling through - ruder (or more intimate depending with whom you're talking).
Keep in mind that context is everything and tone matters so it can be rudder or more pleasant depending on those.
Hope that helps
