You would think that separate and apart are related, just because of how they're spelled. They both have that par in there. It's probably a good way to remember how to spell separate. But guess what? Not all par parts are created equal.1
Apart comes from the French a part, which comes from the Latin ad partem: ad (to) plus partem (part, piece). Think of a single item or piece by itself. If you live in an apartment building, you live in a place where each person's space is kept apart from the others.
The par in separate comes from the Latin parare, which means prepare. It's the se- that means apart! To separate, then, means to pull apart (apart-prepare). That se- also shows up in words like sever (to separate) and secret (set apart, concealed). You'll find the par from separate in words like apparel and parent. That's because parare comes from an older Proto-Indo-European root *pere-2 that means produce.
So apart means put aside and separate means make apart, in a way. When you are deciding which one to use, you can think about whether you mean put to the side or not.
Some people say "created equally," but that's not right. Created equally would mean that the amount of creating that happens is the same. Created equal is short for created to be equal, which is the point.
I use Online Etymology Dictionary for some of my research. They use the asterisk (*) to indicate words that can't be proven in any written source. I've kept it here for accuracy.
Why no examples of "apart" and "separate". I seem to have a book title in my head: A Separate Peace by is it John Knowles? Did he corrode my usage with mordant wordplay?