What is the difference between Refined Storage and Applied Energistics?

Refined Storage is the simpler of the two and was heavily based off of AE1. AE2 has been around for years and was a direct descendent of AE1, while RS is much younger and started mostly scratch with AE (both versions) as inspiration. A lot of people prefer the simplicity of RS, while a similarly large portion of people prefer AE2's complexity due to enjoying the puzzle aspects.

Can I use this mod in my private/public mod pack?

Yes, you don't need to ask permission.