It's really not the perfect metal for neither bullets or missiles.

Tungsten will strip out any barrel, even normal copper jacketed bullets can strip a barrel after a few 1000's of rounds.

Some calibers are prone to stripping more than others which is why some high end rounds will be covered by a polymer coating that adds lubricity and protects the barrel (it also has the benefit of dampening some inconsistencies in the barrel itself which can improve precision).

Tungsten given enough velocity will go through pretty much anything, this isn't something you want with bullets both because you usually do not want bullets to go beyond their targets and that a bullet needs to be able to dump it's energy into the body to be effective.

Small caliber through and throughs leave much less tissue damage unless hitting a vital organ and even then a larger, slower round that deforms would transfer more energy to the target.

The velocity of the bullet has a hard limit and that is the energy that is stored in the chemical propellant.

As for missiles tungsten is heavy as fuck, it's denser than Uranium and Lead you want your missiles to have as long of a range as possible, be as fast as possible (to be harder to intercept) for that you need them to be as light as possible.
If you can make your missiles take on more weight and you've already achieved the desired range and velocity then you better off just putting more stuff that goes boom than an inert metal that does nothing.

Tungsten, just like high carbon steel and depleted uranium is useful for only one thing and that is penetrating armor, in those cases you would optimize your penetrator for that task meaning you'll be using a small narrow profile (basically a dart, or well a flechette) which moves really fast and can compromise heavy armor.

The only real use for tungsten in traditional projectiles it to defeat composite reactive armor that would make shaped charges and other traditional anti armor munitions less or completely in effective.

Even in those cases the tungsten penetrator is more often used to either disable the vehicle by destroying it's engine block, turret controls, suspension and tracks or by compromising the main body armor and then finishing it off with a explosive anti armor around that would also cause crew casualties (in a war of attrition killing the tank crew is often more important than destroying the tank).

It's also important to note that tungsten penetrators are often not made out of pure tungsten since it blunts (relatively) easily, so carbides and other really hard materials would be used to make it even more effective.

Besides that Tungsten might be useful for gravity penetrators basically "bunker busters" although in that case DU and lead might be just as potent unless you are limited by the size of the bomb itself.

Lastly the only other real use of Tungsten is in DIME warheads, these warheads employ inert metal powder as the force delivery mechanism, they fill their effective area with micro-shrapnel that pretty much would tear and liquify all tissue within it's blast radius but due to the high density of the metal and the limited explosive charge have a very small blast radius for intended lethal zone reducing direct collateral damage to zero.