cartridge aka cart is use on a normal atomizer (not a tank atomizer) and is the least popular way to vape
On a manual battery, a drip tip can also be used on a normal atomizer for testing juices and vaping occasional juices or ones that are too thick for you main vaping method. A 510 atomizer uses a 510 drip tip and the wider Ego atomizers use an Ego drip tip.
A cartomizer is a disposable 1-piece combined cart+atomizer (some reservoir types are more like a tank with internal atomizer). Cartomizers are the most popular way to vape favorite juices. They hold more juice than a cart, general need juice added less often, keep your flavors totally separate (not mixed in one atomizer), provide a pretty consistent vape, and no atomizer to keep clean. Downside - bit of a learning curve to fill them properly and in the case of the standard stuffing types to keep them damp so they don't singe. They generally like thinner juices (for basic popular stuffing ones I use 80PG/20VG or less VG and reservoir cartos like up to 30% VG and some 50/50 juices). I also avoid juices the drop a lot of flavoring sediment - that can clog up a carto. The most popular cartos for 510 threading are Boge. I can get 10 days use out of a Boge (which is 3 weeks for me because I use the Boge 1/2 the time and vape another model 1/2 the time). Some people clean and dry cartos to get some additional life out of them, often with inconsistent results.