Snap closures are plastic (or sometimes metal) closures that secure the diaper onto your baby. You can find snaps in multiple places. Some diapers have them across the front, some diapers have snaps as a way to adjust the rise and sizing, and some diapers have hip snaps to prevent wing droop.
Aplix closures can also referred to as hook and loop or velcro (for some reason that was very hard for me to understand in the beginning!). Aplix diapers have a strip of velcro across the front panel with a tab on each side that secures the diaper onto your baby. Most diapers will also come with laundry tabs. Laundry tabs are used to secure the sticky part of the tabs when not in use. This prevents the tabs from snagging onto clothes or getting stuck to other aplix diapers in the wash.
It's up to you which style you choose! I like to have a mix of snap and aplix style diapers in my stash. I prefer snap diapers, but aplix diapers definitely have their place! I like to use them for babysitters and doctor visits. I suggest trying both before making your decision. It's important to note that both snaps and aplix can be replaced on diapers. If a snap pops off, you can contact the manufacturer or even replace them yourself with new snaps and a snap press. Aplix can also be replaced with some simple sewing. CottonBabies (maker of bumGenius diapers) sells refresher kits for their aplix diapers and there are a few online companies that you can send your diapers to get fixed up! Some mamas have even converted aplix diapers to snaps or vice versa!