I've seen that sometimes gods in Greek mythology have different names than the gods in Roman mythology. I've also seen that they are basically the same god just with a different name. What makes Greek and Roman so different that they split it up into two different mythology categories?
Actually, while the evidence is fragmentary and confusing, it points more to their having been separate, and then merging together. There are several elements contributing here to the identification.
There were the interpretatio graeca and also the interpretatio romana in which Greeks and Romans respectively interpreted another culture's gods as their own gods under another name. For instance, Tacitus wrote that the chief god of the Germans was Mercury -- probably Odin. And Greeks so identified Hermes with Thoth that they adopted Thoth's title for "Hermes Trismegistus." Both cultures interpreting the other gods as their own helped draw them together.
The Romans also practiced the rite of evocatio, which was to lure away the tutelary god of a place they were besieging to Rome by offering a greater cult or a richer temple. When the other city was Greek, a Greek god would be added, though they often interpreted this god as equivalent of an existing one as well.
It was never perfect, though. They never succeeded in identifying a Greek god with Janus, and the Roman Mars was considerably better thought of than the Greek Ares, and was a god not only of war but of agriculture.
Romans first worshipped Greek gods because the Greeks colonized the peninnsula first. Later cultures from the Near East, North Africa, the Caucusus and Anatolia had influence. Remember that myth is religion and the differences between Greek, Roman, Egyptian, and any other mythologies are expressed in dogma, belief, and mysticism. Structurally they are essentially the same.
To see the differences look at prescribed social behaviors and taboos. Notice how women are treated, look at how they think about death, war, and afterlife. Look at the relationship of man to the gods. Russian Orthodox and Lutheran are both Christian sects but adherents of one wouldn't last five minutes with other.
They only look the same on the surface but you have to get jiggy with it.