Both! If you can afford it. Most options in this guide feature both a screen protector for your Apple watch as well as a full case to guard the edges and dials, but sometimes the full case will have a screen cover built into it, and other times the case will be screenless with an entirely separate part, and placing a screen cover over the case will be the cherry on top.
At the very least, you will definitely want a basic screen protector covering the expensive display of your Apple Watch to keep it not only scratch-free and drop-proof, but to keep it hygienic also, as most screen protection technology these days boasts antimicrobial properties to fight surface bacteria. If you ever decide to sell your Apple Watch, a scratchless screen will make all the difference to resale value, so keep this in mind too.