There is no such thing as
for apology. You probably wouldn't ever say that anyway even though it's grammatically correct being the negative form of 申し上げる. It is not the meaning you think it is.
What you were asking is probably about
It's a formal apology that you would use after making a mistake in a formal situation i.e. a workplace.
You could use
in informal situations as well. For example, if you caused your friend some trouble somehow, you could tell them 「申し訳ない」.
The distinction between
becomes apparent here:
ごめんなさい is for when you want to apologize for your actions.
すみません is when you express your feelings regarding your mistake.
申し訳ありません is when you want to express your sincerity and regret at your mistake; you're literally at a loss for words regarding your mistake if you say this.
The focus of all these is different. If you done goofed towards your friend or caused them lots of inconvenience, it would be appropriate to say 申し訳ない to them. Similarly you could say 「すまん」.
(This is a good example: http://okwave.jp/qa/q6530648.html )
(By the way, someone else in their answer used the kanji for すみません, but that is an unusual and uncommon thing to do and I highly recommend against it as it looks very out-of-place.)