How to prevent "tufts" in paper when cutting fine details?
A big factor to this would be to make sure your cuts are straight and not at an angle. If your cuts meet on top but are not completely vertical then a tuft would appear on the bottom of the paper, for example. This can be hard to work with since the blade of a craft knife is already at an angle. This leads into one possible solution.

Depending on what you are willing to get away with as far as quality or experience, one surefire way to try and address this is to over-cut your lines just a bit. When trying to meet at an edge or a corner it sounds like you are coming a bit short. That is what is creating the tufts. If you over-cut the line just a bit that should account for that and make the corner or edge clean. Problem with this approach is if done wrong you can cut too far in which can ruin the piece.

You don't even have to do it too much. Just enough to account for my first paragraph. 



I am sure you are already doing this but for the sake of completeness your cutting surface plays a role here. Something simple like a self healing cutting mat will help prevent your paper from moving too much. Sometimes changes in hand pressure can move the paper enough that it starts to pull away from the cut line. A proper mat should help immobilize the work more. 




  Usually I use printer paper or origami paper.


Not sure if this is the best advice, as it could be tedious to consider, but common computer/printer paper has a grain direction. If you look at a common 8 1/2 x 11 you should see that the fibers are lined up from top to bottom. You can also "feel" this when you bow the paper in opposite directions. Folding it from top to bottom has a little more resistance that it does from the left and right sides. 

Point being if you are making cuts it is that second cut that will be an issue. If you can make it so the second cut is the one that cross the grain of the paper it is less likely to tear off. Holding the paper behind the cut can also help mitigate this. 



I cut out a lot of names printed from fancy fonts and this happens to me as well.