The fact is that apart has two meanings:
1- distant in space or time.
2- with the exception of.
However, aside only has meaning 2 but can also mean "to the side of"
Examples:
The houses are 2 miles APART
The conferences are 5 hours APART
APART from her being lovely, she can also be feisty
The dog was standing ASIDE its owner
It was a sunny day ASIDE from the fact it was cold
Of course, there are a few more uses of them both but these are the main ones I can think of :)