Expanding / Hollow Point bullets
Do work better, for many applications. For others, solids (cast, or even FMJ) are preferred. For hunting, game laws specify what kinds of bullets are allowable. Generally, expanding bullets of some type are required, and for sporting purposes, cast lead is also considered "expanding", even if the reality is somewhat different. Full Metal Jacket bullets are (generally) not allowed for hunting.
As others have explained, hollow points expand the area of bullet effect, and so, are more effective than a non-expanding bullet in the same spot. In simplest terms, when a hollowpoint expands, it becomes in effect, a bigger bullet.
Many police departments switched from round nose lead to hollowpoints because there were fewer fatal shootings and more successful perpetrator stops than before.
Actually, this is only partly true.
While the fact that using hollowpoints often means that fewer shots are needed to stop an adversary, (and the fewer non-lethal holes in a person, the more likely modern medicine is able to preserve their life), this fact is not the reason the change was made. Nor was officer safety and preservation the driving force either, although that is also a benefit of hollow point use.
The change to using hollowpoints came about finally, because of one thing. Politics. And by that, I mean the politics of perception, within police management. The hollowpoint bullet is not a new thing. Not even remotely. For more than the past century, police
could have been using hollow points, yet they did not. Because they were not allowed to, by their political masters, the police chiefs. Police chiefs/mayors/etc. did not allow their officers to carry hollow points, due to their belief that the public perception of the police using hollowpoints would be that the police were "killers".
Officers were "always" lost during the performance of their duty, it was one of the recognised, and accepted risks of the job. To those making the decision, arguments from the officers about increased effectiveness of hollowpoints did not carry as much weight as their fears of unpopular public reaction. And they held this view for genrations, because the public could see the ammo the officers carried.
Back when virtually all police carried revolvers, their spare ammo was carried in loops on their belts, open to public view. Even after the police switched over to dump pouches or speedloaders for their carry ammo the attitude of "no hollowpoints, they scare the public" persisted for a long time. It wasn't until police began switching over to semiauto pistols (where the ammo in the spare magazines could not be seen) that this entrenched attitude on the part of police
managment finally went away.
Once this roadblock was finally overcome, and hollowpoints became general use, their benefits in officer safety, and survivability became widely recognised, and the public's (and police management's) attitude has changed, as reality finally sinks in. It could have been done generations ago, and many fine officers who are gone would likely have not had their lives cut short, but politics (including internal police politics) always seems to lag behind reality at least somewhat.
It is a good thing that the attitude today is so much different, with officer safety and effectivness being much more important to the public, and the police policy makers.
Air Marshals carry a very highly frangible hollow point bullet when flying. The reason is so if they have to shoot in the aircraft and for some reason miss, they do not puncture the aircraft skin and cause depressurization.
They do carry frangible ammo, but the myth of a bullet hole (or several) causing depressurization is just that, a myth. An important plot device in the James Bond movie
Goldfinger, but a hollywood myth, just like so many others.
Blowing out a window in a jet airliner will cause a loss of pressure, and the oxygen masks will drop down, but it will not (and cannot) cause people to be sucked out of the plane, like one sees in the movies. It is not like opening the airlock on a spaceship. Planes are pressurised, but the difference between inside and outside is not very much, comparatively. And the pressure inside changes (notice how your ears pop?) airplanes leak air all the time. It supposed to be that way. A number of years back, one airliner lost 12 FEET of cabin roof! One person, a stewardess, who happened to be standing directly underneath, was sucked out. No one else was, and the plane did not crash!
The real reason Air Marshals carry frangible ammo is not that there are worries about shooting through the skin of the plane, but worries about shooting through the hijacker/terrorist/bad guy, seats/cabin walls, and hitting the pilot or cockpit controls, or other innocent passengers.