I will be the first to admit I am not terribly well versed in Roman Religion, or at least it has been quite some time since I looked at it, but I doubt that fact is of huge importance to this subject because Apollo was altogether directly borrowed from the Hellenes. The Romans simply didn’t have any god that they considered the equivalent of what Apollon was. Thus all you see as a modification of the name in a Latinized form…and really a very slight modification at that. This is unlike other deities whose myths and attributes were conflated by the Romans with their own gods, which usually produced cults that had distinct differences from the Hellenic ones, but this was simply not the case with Apollo. Unlike the relationship Apollon enjoys with Helios, I have never particular come across anything like that in regards to the Roman Sol. The Roman philosopher in his book On the Nature of the Gods, does says that Apollo is the Greek name of Sol, but I hadn’t previously come across this statement elsewhere and as Apollo’s temple was not part of the city proper, as where would be any gods native to Romans, it seems like a dubious statement in regards to actual belief and practice. There was one scholar I had read some time ago who said that the Romans gave little attention to the features of Apollon’s cult which resembled features already in place with their own native god Sol (my paraphrase). I have no idea how accurate that particular idea may be though it is my general impression that the Romans didn’t make a particularly strong association between the sun and Apollo in their cultus to him, given the Latin epithets that we have don’t even address light much less a specific light-bearing body.
However, he was at one point associated with Luciferos who, though a light bearing god, has no association with the kind of solar light and was rather associated with the morning star, perhaps inspired by Apollon’s epithet Phosophoros. That Lucifer/Phosophoros was connected with the evening star and philosophical statements stated that the evening and morning star were the same it may allude to how the Hellenes and in turn the Romans perceived part of the relationship with the sun, as the star followed before the dawn and after the sunset in the twilight hours which are particular to Apollon Lykeios. That is to say that a recognition of this period between night and day, during the twilight hours, in which the planet Venus shines the most brilliantly. Whether or not either the Hellenes or Romans believed Phosophoros or Luciferos to be directly Apollon or Apollo is debatable, but the concept of the light as being in a related dance with the sun but not being the sun seems to be a common expression in both cases. However, it is possible that at least with the Romans they may have taken a more direct association between the morning/evening star and Apollon, if we consider Diana whom they conflated with Artemis and to whom they attached all of the latter’s myths and symbols. Not only was Diana called Lucifera, but also the philosopher Cicero attributes a particular manifestation of Diana as moving through the heavens as a wandering star. However we do find some divergence artistically as they seemed to have incorporated many elements of a solar idea, more to the point in expressing Apollo’s connection with the sun to illustrate the passage of time or as a vague kind of solar cross halo behind his head in mosaic portraits of the god. Typically though most images of Apollo were, like so many statues, copies from those made by Hellenes. The exception being Etruscan work, such as the statues which made the scene of Herakles and Apollon of Veii contesting over the golden hind. The Etruscan Aplu, or Apulu, is thought to have been borrowed from the Hellenes at a much earlier date and while he never had much presence in Etruscan myths and cosmology he had a popular cult with numerous votive statues among the Etruscans.
From my understanding much of the nuances related to Apollon’s domain among the Hellenes were pretty much stripped away down to a more simple presentation of the god, and one that probably colors how people perceive the Hellenic god today rather than embracing his complexities. There is no evidence of Apollo carrying over any of the characteristics of destruction, death and tombs that Apollon has, as well as his involvement with orchards, roads, pastures, herds, etc. Instead, we find Cicero dealing with some of these parts of Apollon’s domain as entirely separate gods.If that is how the Romans perceived them philosophically then that may explain why so many various functions of Apollon get neglected. That is to say that aside from Latinizing his epithet Phoibos into Phoebus and bringing it into use during the Imperial period, his epithets among the Romans are few and rather straight forward. These are: Articenens (who carries the bow), Averruncus (who turns away evil), Coelispex (who watches the heavens) which is most likely an oracular/divination association, Medicus (doctor), and Culicarius (who drives away midges, a small two winged fly). Thus you have Apollo as an archer, oracular god, a deity that turns away unpleasant and evil things, and a healer (the latter associated with the former if you consider turning away disease in association). In the later case the Vestal Virgins were known to pray to Apollo Medicus by chanting his name. Apollo Medicus in association with the Vestals is perhaps the most well known forms of Apollo in Rome. It was primarily as a healing god that it has been speculated that he was brought into Rome and until the Imperial period during which Augustus identified himself with Apollo and brought his worship to greater prestige, including the building of the temple on the Palentine hill…outside of the original city limits as befits a foreign god.
It was mostly out of the interest of this emperor that Apollo was established further in recognition and popularity in Rome from what I can tell, who then instituted the gods place on the Palentine Hill separate from the Apollinare on the Flaminian fields. This of course should be distinguished from any temples that may have existed in southern Italy that were founded by Hellenic colonists. This is in a huge contrast to the numerous temples of Apollon of great antiquity scattered through Hellas and Ionia which is quite telling of how important this god was perceived in the two different cultures. Likewise, as far as I have come across the Romans only had one event that was especially for Apollo, the Ludi Apollinares, a series of games in honor of the god to celebrate his coming to Rome and relieving the people of the plague. This is in stark contrast to the numerous festivals held for Apollon throughout Hellas that honor mythical events and specific functions of Apollon’s domain rather than an event. However, poems seem to be largely inspired by Hellenic hymns, though the Roman ones seem to focus more on Apollo in association with transition of boys into adulthood than anything else from what I recall. Though many of the poems otherwise have a carry over from Hellenic poems and myths to the god. But the most celebrated of poems focused primarily on the successful maturation of boys.
As far as myths regarding Apollo, these were directly borrowed from the Hellenes though often retold in modified forms. For instance it is from the Romans that we find the instance of the death of Orion being attributed to a passion Apollon held for Artemis. This is a version of the myth that for all I can tell is distinctly Roman. Likewise the most well known version of the transformation of Daphne is from the Roman poet Ovid, and this is the only version in which I have read of Daphne being struck by a lead dart. In Hellenic versions of the myth she, a chaste maiden, runs as natural instinct would bid her and needs no prompting from a dart. In fact the whole lead dart thing I find utterly silly, especially considering that Daphne became his most sacred medium as his sacred tree. Seems hardly right for a myth to suggest that she was repelled by him by some negative juju. These are just a couple of examples, but certainly there are many others out there.
Another really distinct difference between the Roman Apollo and the Hellenic Apollon is that the former did not possess a domestic cultus. In Hellas Apollon is well connected the household as a protective deity in the form of a black phallic stone before the household entrance. This same image has been found on a plaque from the Palentinian temple but aside from a scene in which a youth and maiden are honoring the phallic stone draped with arrow and lyre to clarify its association with the god, there is no evidence of Apollon being associated with the roman household in this manner. Apollon has no presence as god before the doorway, lord of the gate. Likewise the beginning of the month, an important occasion among Hellenic households, did not honor Apollon Noumenios or any form of Apollo associated with the new month. Rather the new month was auspicious to Iuno. This completely alienates their Apollo from the most primary protections he provides, the protection of the household, which leads me to think that the protective element of Apollon’s nature was possible either ignored and given less importance. Of course Romans had many of their own gods, numerous actually, associated with every part of the house. That said, it seems strange that this form of Apollon that is so vitally important in the daily life gets forgotten aside from a small plaque on the Palentine temple.
In short I would say that Apollo in the end is more like Apollon-light, a watered-down version of Apollon that was suitable for the Roman needs, and as such is perfectly fine, but lacks the beauty, complexity and depth that Apollon contains within himself in my opinion. That is not to say that during the Imperial period that Apollon’s cult didn’t enjoy a great expression of beauty in Rome, but in the end it just doesn’t come across as being quite close enough to Apollon in Hellas.