FMJs are primarily to avoid any feeding problems. These bullets have a rounded nose and hard jacket material that helps prevent any hang-ups upon feeding. It is also a requirement for military use––-international law requires FMJ for war use.
Flat points do a couple of things. One, they increase penetration in soft tissue, and harder things like wood. They also bounce back a bit better so it makes it easier to knock-down heavy steel targets. Wad cutters make nice clean holes that are easy to score. This is why they are used in target ammo. Semi-wad cutters do the same thing, but have less air resistance, thus more velocity at longer ranges. Semi-wad cutters also make very good hunting bullets. With the right alloy, they will expand in game, but also have good penetration because of the flat point.
Hollow points are made to expand in soft tissue, thus making a larger wound channel, and the hollow point also reduces penetration. When a bullet expands, it takes energy to do so. The energy comes from it's mass and speed. It must slow down upon expanding, thus it won't go as far into the animal or person.
The notches on hollowpoints are a way to control how the bullet will expand. The jacket will start tearing at the notches and will open-up like a flower with a petal between each notch. Hollow point bullets that don't have notches might expand in other than nice uniform petals. Unseen from the outside, however, many of these bullets do have notches or other methods to get the bullet to expand in a controlled way. You will also find that the 185 grain .45 bullets have a larger cavity in the nose than bullets for faster calibers like the 9mm. This is because it more difficult to make a bullet expand at lower velocities. Hollow-points fill with fluid and burst like a pipe. Thinner pipes are easier to burst.
Soft point handgun bullets are principlally used for larger game animals where you need penetration, but with a bit of expansion as well. Without the hollow-point, the bullet will not expand right away, and if the striking velocity is too low, it won't expand at all.
BTW, it is counter intuitive to think that a bullet with a flat point will penetrate further than one with a sharp point or even a round nose. What happens with the pointed bullet is that tissue is pushed to the side but grips the side of the bullet as it travels on. The flat point cuts the tissue which make for less to rub on the sides of the bullet.