There are plenty of ways to classify paint gun types, but in the most basic sense it comes down to cocking action. On pump guns the cocking action is completely manual - in other words, you pump the gun once before each shot. Once upon a time in paintball, you really didn't have much choice about using a pump gun, simply because nothing else was available. Even after reliable semi-automatics entered the scene, pumps were still, because of price, considered entry level guns. However, two things largely changed that: the debut and success of low-cost semi-automatics like the Kingman Spyder, and the influx of reliable semi-automatics into the rental market like the Tippmann Pro-lite. Now semi-automatics are considered entry level equipment and pumps are considered mostly for the extremely budget minded or for the traditionalist.
There are various opinions about the use of pump guns. On the one hand, some players wouldn't consider giving up their competitive edge by using an archaic pump style gun, while traditionalists using pumps paint those who don't as equipment-dependent wimps. Regarding equipment for new players, one school of thought advises against new players using a pump out of fear the new player will be overwhelmed, but another insists that playing pump is the best way to aquire the skills needed to be successful. To be honest, I can see truth in all of these viewpoints. However, I think whether or not someone will enjoy using a pump basically comes down to each individual's personality - in other words, there is no true blanket statement that fits all players on this issue.
In a field of semi-autos, a pump player definitely has a disadvantage. Some pumps, when properly set-up, can put out an impressive rate of fire, even matching some entry level semi-automatics. However, the biggest single disadvantage of using the pump is that the action requires two hands. With a semi-automatic, it is much easier to shoot while crawling, running, or hanging out from a bunker. If a pump player wants to be successful in a field of semi-autos, he/she will have to work harder.
There is the general perception that players use less paint while playing pump. While it's true that many people say they play pump for that exact reason, the fact is, that's a tactical decision. I have personally fired in excess of 400 rounds in a 20 minute game while using a pump. If using less paint is your goal, that can be done with the judicious use of your trigger finger, regardless of the gun.
For years, pumps were considered to have better range and accuracy than their semi-auto counterparts. Actually, for years, this was true. Compared to such guns as the Tippmann 68 Special, and F1 Illustrator, pump guns, as a whole, shot much better. Semi-automatic technology was new, barrel technology was in its infancy. That's why many of us stuck with pumps even after the new semis hit the market. However, things change. Today's better semi-automatics are as capable as pump guns ever were - even more capable when you figure in ease of operation and rate-of-fire.
So why use a pump at all? Well, this is where the personality part comes in. Personally, I find it more challenging to use a pump, and I like a challenge. In most situations where I choose to use a pump (and I am by no means a "pump only" player), it usually has more to do with my mood at the particular moment than anything. I know up front that I'll have to play better, and have to put more effort into my game - and most of the time I do exactly that, as if some switch inside me were flipped on. I also know if I am successful there will be a sense of satisfaction in having met the challenge I set for myself. The concept of making things more challenging for myself is all about me. That's why I say it depends on your personality. Do you want to take the challenge?