Here's the big difference in the mini kits to me.
KR808-D1 typically runs around 4v under load. The cartos for it are horizontal coil and typically run around 3 Ohms. A full KR-808D-1 cartomizer set up 'stock' and running off of a 'battery' will usually last a few hundred puffs before you need to replace/refill it. You can easily add an 808>510 adapter and use the 510 stuff on it if you want it to feel a little warmer. You can also run a 901 atomizer on it if you want to drip or pack carts (very small carts albeit). If you want, there is a plethora of prefilled cartomizers available for the 808 at the same price as the blanks...so if you find a preloaded flavor you like, you can vape them dry then refill with your own flavors several times (and get a little liquid for free prepacked)
510 mini kits use a battery that typically runs around 3.5ish volts under load. Those kits usually come with a pair of atomizers and 'carts', but you can also pick up a five pack of 510 Cartomizers for it. Standard atomizers and cartomizers for these kits usually run at 2.5 Ohms of resistance, and it's safe to drop to 2 Ohms for a little more heat (but will drain the battery a little faster).
The 510 atties/cartridges don't hold very much. The 510 cartomizers are usually a vertical coil design instead of the horizontal type. They do perform well, but typically need topping with liquid a little more often than the horizontal type coils of the 808.
I personally go for the 808 batteries and run short 510 CE2 cartomizers @ 2.5 Ohms on them for a little hotter vape. When I really want a true mini PV.
Just for info purposes:
Cartomizers put the cartridge (holds the liquid) and atomizer (coil that heats the liquid) into one piece.
Atomizers have no polyfill in them for holding liquid. You either direct drip onto the atomizer, or push in a cartridge that holds the liquid.
510, 808, and 901 sized cartridges typically perform poorly. They're just too small, so unless you mod them up with alternate fillers (blue foam, tea bags, etc), they're kind of lame. So many people go for the cartomizers, or move up to bigger platforms (Like the 801, LEO, Mega 510) if they want to pack carts from time to time.
Better yet...
Take a look at the 801 pen style or LEO units!
The 801 is probably the most underrated off the shelf e-cig kit there is.
Here's what rocks about the 801.
The stock atomizer for these is amazing...it's over-sized, and the native cartridges for it work really well (no need to pack them with blue foam or tea bags) and hold quite a lot of fluid. 801/302 cartomizers will also fit and are superb performers! You can also drip in these over-sized atomizers if you like. In short, you get an atomizer system that performs as well as some of the best premium cartos out there. It works well at lower voltages. It holds a lot of liquid. It's going to be in a similar price range to either the 510 or 808 kit. You can easily find atomizers in this type from 1.7 Ohms (hotter vape) on up to 5 Ohm HV gear (Cooler vape...you'll probably want to stay at 2.5 Ohms or below for the stock 801 batteries). They do use 'whistle tip' type carts...that's the only thing I can think of that might be a turn off for some people who 'hate' whistle tips. They perform as well or better than Mega 510 atties, and cost half as much

These are slightly bigger than 808 and 510 units (mainly because of the atty), but aren't as girthy as an eGo. Drop it in your shirt pocket and it looks more like an ink pen.