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.)