A pop-up drain screws into the drain by means of a threaded screw in the center of the plug. Turn the plug one way to lower it and seal it to the drain. Turn the plug the other way and it pops up and allows water to flow out of the tub. Turn the plug too far and it comes out and you have to screw it back into the tub tediously to the proper level. These drains work well and are easy to keep up with, but they can be difficult to clean and are easily fouled with hair.
A trip lever drain plug is connected to a lever, usually on the top of the faucet. Lift the lever and it lowers the drain plug and seals it over the drain. Lower the lever and the plug lifts, allowing water to drain out. These types of drains are simple to operate, but if they cease working, they can be complicated to fix, since reaching the lever mechanism requires removing the faucet and drain plug. Some trip lever plugs have more clearance that others. They can become clogged up with hair and soap scum and may be difficult to clean.
A plug drain has no complicated moving parts, You simply block the drain with a plastic plug, much as a cork stops up a bottle. The plug is easy to clean, since you can scrub all the way around it. Properly sized, the plug does a good job of keeping water in the tub. The chief disadvantage of the plug is that they're easy to lose. You can attach the plug to the faucet by means of a chain, but chains can wear out or the plug can come loose.
The type of drain you use may depend on the type of bathtub. Older, claw-foot tubs are best suited for simple rubber or plastic plugs. If your tub has a trip lever, you can replace it with another trip lever or a plug, but you'd have to remove the old drain altogether to install a pop-up drain. The opening left from a pop-up might be too wide to accommodate a simple plug.