Sol is just the latin word for the Sun, as Helios is just the Greek one. A Temple to Sol is just a temple to the sun. Apollo would be the dominant Sun god of the early Roman Empire but there would have been other sun cults completely lost to history.
Mithras is a very specific deity with a Persian based name in a very specific cult/secret society during the later Roman Empire. You can read about what little we know about that cult in a lot of recent books but there's also a lot of misinformation caused by 19th century speculation.
Sol Invictus belongs to the same era as Mithras but I've never heard anything concrete about him having a organised cult anything like Mithras. From how paganism tended to work I assume he was thought of as a specific aspect of the sun and therefore would overlap with but also be worshipped distinctly from other solar deities. But really I'm thinking of the Egyptians with their diverse and numerous aspects of Ra. I wouldn't be surprised if a time traveler asking your question wouldn't be able to find one Roman who said Apollo, Sol Invictus and Sol were just different names for the same thing and another Roman who would vehemently disagree.
Mithraism was very stratified with ranks and initiatory levels. The impression I get is that Sol Invictus worship was a lot more open and inclusive while still being most popular with the same soldier crowd who liked Mithras and Jesus.
Sol Invictus is also influenced by the Imperial Cult and numerous attempts by Roman Emperors to connect their rule to a Sun god such as the Nero Colossus that the Collesium is named after. As far as I know Mithras was never so closely associated with any Emperor but Sol Invictus was invoked as a personal protector by Emperor Aurelian.
Art wise, Sol Invictus just looks like a normal Greco-Roman sun god and can't be clearly differentiated from the Hellios of centuries earlier. Mithras on the other hand wears a Phrygian cap which is a motif associated with the east and foreigness. Some artwork shows Mithras and Sol Invictus as seperate characters in the same scene but polytheism is messy and syncretism was dominant in Roman religion.
Polytheistic gods aren't Pokemon, the encyclopedia approach to classifying them isn't especially helpful. They had a tendency to blur into each other and were constantly being combined in novel ways for particular purposes. There's a tendency to for example take one story about Sekhmet and Hathor turning into each other as some kind of fact about one being the permanent aspect of the other but there's enough contradictory information produced by that approach that I would strongly discourage reading any single source as a definitive statement of any kind of doctrine.
At least one source has Sol Invictus being a name of Mithras, but personally I would read that as only saying that the writer chose to make that conflation in that moment for that source. If Mithras and Sol Invictus are to be differentiated I would do so by form of worship not by speculating on the character of the supposed gods themselves. - ACable89
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