they're both the same, however if i was to try and seperate the two i would say that "with the idea" is expressing your concern for leaving your friend/girlfriend (slightly protective and worried). The second one is just more direct and less information is provided to the listener, it would be more passive/upfront.

The first sentence is stating what has not happened yet, it is just an idea and your not comfortable with it. Maybe something you thought about doing but are doubtful of doing. The second sentence is real time and your actually leaving her on her own and your not comfortable with it. You write the sentence as , l am not comfortable leaving her on her own. Leave the (with) out. Hope this helps.