The founding of Rome by the twins Romulus/Remus is but one part of the legend, the other being the prehistory of the founding of Rome under Aeneas. The story of the founding can thus be divided into three parts: the coming of Aeneas to Latium after the fall of Troy, founding a settlement called Lavinum; the founding of Alba Longa thirty years later by Aeneas' son Ascanius/Julius (this bit became more popular under Julius Caesar since his house claimed to be the descendants of Julius); the founding of Rome itself three hundred years later by a descendant of Ascanius/Julius. This legend however did not always form one whole. Rather, they were both independent attempts at explaining the same phenomenon - the founding of Rome. The story of Romulus was Latin in origin, while the story of Aeneas was Greek.

Later sources such as Servius and Dionysius of Halicarnassus allow us to trace a pattern in the growth of the legend. As far back as the 5th century BC, the founding of the city was later ascribed to an eponymous 'Romulus,' who was later associated with the god Quirinus. Likewise, in the fifth century, writers such as Agathocles of Cyzicum attributed the founding to another character named 'Rhomos.' Others, such as Hellacinus of Lesbos, states that Rome was founded by Aeneas, naming it after a Trojan woman called 'Rhome.' Timaeus of Tauromenium states that it was founded by a man called 'Rhomos,' who was the grandson of Aeneas.

At some point, the Greeks became aware of the Latin Romulus, who 'existed' simultaneously with the Greek 'Rhomos.' Alcimus the Sicilian states that Rhomos was the one who founded Rome, but was the grandson of Romulus, who was the son of Aeneas. Callias of Syracuse states that a trojan woman named Rhome married a Latin king and had a son named Romulus, who founded a city in her name.

As you can see these accounts are quite awkward and confused, and the Romans must have realised that they needed a concrete foundation myth to match those of other rival cities (such as Praeneste, whose foundation myth was traced to Ulysses and Caeculus). It is unclear when the Romans decided to fuse the Aeneas/Romulus traditions, however. The Greek writers did not account for the 400 year gap between the landing of Aeneas and the founding of Rome, so it is likely the Romans created the list of kings of Abla Long to bridge the gap. The Latins also transformed the awkward Greek 'Rhomos' to the Roman-Etruscan 'Remus,' and seized upon the Greek literary trope of twin rivalry (Romans loved to borrow all manner of Greek aspects; the twin rivalry thing can be seen in cases such as those of Pelias and Neleus or Castor and Pollux). It's unknown why Remus had to be killed - maybe as a symbolism of shaking off Etruscan influence, since Remus was an Etruscan name. Or maybe it was just to simplify the myth and concentrate the importance upon one dominant figure.

Regardless, the greatest evidence of syncretism between the two legends can be seen in Vergil's Aeneid, which is very much the main story for the founding of Rome. As seen, however, the unitary, coherent foundation myth was not always so, having to incorporate other elements from various Greek versions of the same legend, and then altering them throughout the centuries.

Sources:

Brinkman, J. A., “The Foundation Legends in Vergil,” The Classical Journal, vol. 54, no. 1 (1958), pp. 25-33.

Donlan, Walter, “The Foundation Legends of Rome: An Example of Dynamic Process,” The Classical World, vol. 64, no. 4 (1970), pp. 109-114.