In the simplest way what's the difference between 对不起 and 对不起嘛?
The 嘛 here has actually no real meaning but just a modal particle. 

In Chinese culture, there is a word 撒娇, which is not easy to find the direct translation in English. This happens mainly on girls, who would like to act like tender and sensitive to some intimate friends or partners. With 嘛 the young woman could transfer this additional meaning to others. If a boy act like this, he would be said girlish.

This is widely used not only in Taiwanese dialect, but also in other dialects from mainland.