“Are they the same thing, or is that some birds have beaks (pointed mouths) and others have bills (rounded mouths)?”
A beak and a bill are actually the same thing! These are just two different terms for the same part of the bird.
However, there are some unscientific but common terms used to describe birds using the word “bill” and “beak” to differentiate between species/breeds.
For instance, the term “Softbill” is used for frugivores (fruit eating birds). However, not all “Softbills” have soft bills! Nor do they all eat fruit. Softbills do tend to have similar shaped bills (smaller, rounded). But this doesn’t mean they are not beaks. Or that all Softbill’s have that shape.
Another example is “hooked beak” to describe larger, nut and seed eating parrots. In this way there is a difference, but its only a colloquial (and somewhat vague) form of identifying large groups of birds. If you talked to an avian expert or vet, they would tell you that the term beak and bill is interchangeable.