Didn't read the replies so I don't know if this was already mentioned. The two names are not the same. They are not only spelled differently but they are also pronounced differently.
Amelia is uh-meal-ya.
Emilia is A-meal-ya.
I am not picky about nn's...I think if you like Emmy, go with Emmy for either name, but pick the name you want with the correct pronunciation in mind.
These names with same nn is also on my list...I went to a website that pronounces the names for you to hear. I do like both, but Amelia more.