The historiography of the Romulus-Remus myth is extremely rich and interesting - in my answer, I'm going to look at some of the questions and approaches that people have used to approach the mythical brothers and the question of founding of Rome in general.

I think the most basic question one can go and ask is whether Romulus and Remus were real. Already here, you can get a pretty stark difference of opinions among historians: Andrea Carandini, an Italian archaeologist, is famously convinced that Romulus was an actual historical figure, and has led excavations on the Palatine that turned up what he claims is a Romulan-era palace and walls (it's not). On the other hand, the vast majority of British classists would very strongly disagree: and you'll get a range of opinions there, from T. P. Wiseman (who argued that the myth is a fourth-century construction reflecting plebeian ideology) to T. J. Cornell, who will very cautiously say that even though Romulus and Remus are a myth, it's a very early one, and will point you to stuff like the Bolsena mirror, which portrays what some people would say is an early depiction of the twins.

So why people can't agree on something what arguably a very basic question? One part of it has to be that we don't even know what we should look for: even though one usually refers to 'the myth of Romulus and Remus', Cornell points out it is anything but 'a myth': our assembled antiquarian sources present us with no less than twenty-five distinct versions of the foundation story. In some of them, Remus lives to a ripe old age, and in some, there isn't a single mention of the brothers (most importantly, note the competing idea that Rome was founded by Aeneas - we will come back to it): so already when you are trying to prove that the myth is real or not, you are picking parts to assemble what 'the myth' really is. Another problem would be, of course, the nature of our evidence: for written sources, we mostly rely on Livy and Dionysus of Halicarnassus (both writing roughly in the 1st century BC). They in turn supposedly rely on historiographical tradition going back to the 3rd century BC, working with stuff like the first history of Fabius Pictor, Greek historians and assorted pieces of history like the list of consuls, which might or might not include actual antiquarian data (it's hard to say whether they actually work with these sources - Livy in particular gets called out quite a bit for just getting things wrong and mixing stories together). At best, we still only know what people in 3rd century BC said about Romulus and Remus (traditionally dated to 8th c. BC!), and at worst, we have a couple of Augustan retellings that tell us absolutely nothing about the twins or the founding of Rome. In addition to this, we have a couple of extremely interesting but awfully cryptic artifacts: there is the Bolsena mirror, which might portray the twins or the Lares, an 'Aeneas' inscription from Tor Tignosa which might be talking about Aeneas or perhaps a woman called Aula Venia, or the Capitoline wolf - historians are still not sure whether it's from 5th c. BC or 11th c. CE.

In addition, there is the fact that the Romulus and Remus legend is extremely odd: you have two founders of a city (what do you need the second one for?), the Aeneas/Greek element that gets tangled into it (typically in saying that Aeneas is the ancestor of the twins - this is extremely weird in comparison to Greek foundation myths, which emphatise the native element of the people of the city - the idea that they were born from the earth or in any other way were 'always there') and the fact that one murders the other. Already the Romans were pretty confused by it and debated it quite heavily: the idea that the 'lupa' in the tale was actually a prostitute is already recorded in Livy.

So what do we get out of it? Well, what I would argue comes out extremely clearly is that the 'myth' of Romulus and Remus is precisely that - a myth. What we have is a bunch of stories people told each other about the origin of their city (Mary Beard points out that the name of Romulus literally can translate as 'Mr Rome'), used to explain where they came from: they got synthetised and retold until historians like Livy wrote them down to the best of their ability, mixing together stories they've read and known and used them to make a point - for instance, the violence of the story (Romulus comitting fratricide) likely got ramped up in the Late Republic, as people tried to explain the murderous politics of the era by saying that civic strife and conflict was literally written into the 'DNA' of their state. The Greek element, on the other hand, probably started with Greek historians, who were keen to explain the rise of Rome on their own terms (by arguing that Rome was founded by Greeks, and therefore was actually sort-of Greek and not just a bunch of jumped-up barbarians that somehow took over the Mediterranean) and got taken up by Romans, who were equally happy to write themselves into an ancient lineage that gave them a 'rightful' place in the Meditteranean world and cosmology.

So should we care about Romulus and Remus at all? Well, the myth not useful at all for discovering the origins of Rome: but it is extremely interesting for reading the politics and identity of the later state. By looking at how it was deployed by later authors, we can see how people like Cicero (In Cat I.) saw their own state and era reflected in the past, the perception of which they themselves shaped: and it can help us realise the importance of history as embedded in the language these people used to talk about themselves and their politics or identity. It's extremely important that Augustus thought about using the name 'Romulus' instead, and ended up not using it: it's equally important that as he rebuilt the Palatine, he took care to preserve the supposed 'Hut of Romulus' that apparently stood there from the time immemorial. The Romans thought about Romulus and Remus, expressed themselves throughout in the terms of the story and remembered it in the landscape of the city as well as in their collective 'historical' and political memory. The myth mattered to the Romans, and this is why it should matter to us.

For further reading, I would recommed M. Beard's SPQR, chapter 2 and the appropriate chapters from T. J. Cornell's The Beginnings of Rome and K. Lomas' Rise of Rome. Primary sources would be Roman Antiquities from Dionysus of Halicarnassus and Livy, books 1 and 2.