Throwing bullpens in practice will not simulate the duties you have on the mound in the game as a fielder. There are a number of fielding situations if not executed correctly, could cost your team a win. Once you release the ball you are the closest fielder to the hitter so you have to be in a position to defend yourself against line drives up the middle and field your position.
Come backers up the middle can have a big impact on the momentum of the game. Say you are pitching late in a tie game, with a runner on first base, and one out. If you are not in a good fielding position a one hopper up the middle will normally result in runners on first and third with one out. If you are in a good fielding position a one hopper back to you will generally result in an inning ending double play.
Fielding bunts is another game variable that can't be reproduced during bullpen work. Generally sacrifice bunts and bunts for a hit happen in close or low scoring games, so errors can be especially costly. An unbalanced followthrough can make getting to and fielding bunts much more difficult as well.
Covering first on ground balls to the right side of the infield is used more often when the pitcher's mound is 60.5 feet from the plate and the bases are 90 feet apart. But even young pitchers can develop good habits during games by hustling towards first on all ground balls to the right side of the field. The most important part of covering first base is getting a good jump off the mound.
Backing up bases isn't physically hard to do, but in the heat of battle when you are dealing with runners circling the bases while giving up hits and runs it can be harder to make yourself get into position. This game variable ties in closely with body language. If you have bad body language, sulk, pout, or start arguing with an umpire you are probably missing your assignment and giving the opponent a better chase to get extra bases. Not to mention it makes you and your team look terrible.
A runner on third in a close game can sometimes make a difference when selecting pitches to throw during the sequence. If you have the confidence in your catcher to throw off speed pitches in the dirt with a runner on third and the ball gets away, you have to be ready to point out the ball, cover home without blocking the plate, catch the throw, and apply the tag. Almost any legal slide will take your legs out if you move too far across the plate, practice this for your safety.
Whenever you pick a runner off the bases you will likely be a part of the rundown that follows. You have to understand which base to go, how to close the distance toward the runner, get out of the baseline when you don't have the ball, catch, throw, and apply the tag. These are all important skills that are not learned in the bullpen.