As you can figure out from my nick and avatar, I am kinda a fan of both. Anyways...
There are obvious broad similarities between the both of them, since they are both based on a mix of Ango-Saxon, Celtic, English and Germanic legends. However, there are quite some more parallels.
Martin himself has stated that he is aiming for LotR-esque "bittersweet ending":
In fact, there are some obvious parallels from the very beginning. Both A Lord of the Rings and A Song of Ice and Fire begin with rumors of a dark power rising: Sauron has not yet obviously returned to power at the time when LotR books start, and it is even more obvious with the Hobbit and Necromancer of Dol Guldur, who is relegated to a side-"story never shown", yet turns out to be the primary antagonist all along. Similarly, in ASoIaF, very few characters believe that the Others are returning to power, and focus is initially on inner conflicts.
As this article points out, Daenerys is Gollum. Now, this more obviously applies to the show. However, the Iron Throne is the Westerosi equivalent of the One Ring in Lord of the Rings: they are both artefacts which represent power, and human obsession with it. Daenerys is fixated on the Iron Throne much like Smeagol is on the Ring. Smeagol is initially a not-that-bad of a person, but Ring turns him despicable; likewise, Daenerys tries to be a decent person, but gives in to obsession with power (this is true in the books as well; it is not a show-only development). Frodo and Jon are both unwitting heroes, people who become heroes by chance rather than by desire.
Martin outright states that he drew lessons from Tolkien on handling of magic: it is there, but it is extremely subtle, so subtle that oftentimes you cannot even notice it unless it is explicitly pointed out. Valyrian Steel weapons are obviously magical, but up until the clash with the Others / White Walkers, nobody would ever take them for anything other than just high-quality steel. This parallels Lord of the Rings, where the Daggers of Westernesse have no magical characteristics except for their ability to kill Ringwraiths. Other usages are also quite similar and similarly low-key: Melisandre can see future events in fire, much like Galadriel can in her mirror; but neither are reliable, and both are dangerous as a guide for future events. Had Sam returned to Shire after seeing a vision in Galadriel's mirror, Middle Earth would have been doomed; likewise, Melisandre misinterpreted her visions gained from the fire, but unlike Sam she acted on them.
Even narrative structure is similar (from the link):
Main difference is that Tolkien's narrative is outward-focused, while Martin's is inward-focused. What do I mean by that? Tolkien always has an outside threat, a Big Bad: Morgoth, then Sauron, then Saruman and Sauron. Primary threat, and thus primary focus of the narrative, is externalized to protagonists' political landscape, even though it is internalized when it comes to protagonists themselves. Sauron is an outside invader; if Gondor is to be compared to Byzantine Empire, Mordor would be Seljuk or Ottoman Empire. From the First age to the Third, the focus is always on defending against the attack from without. Martin's world is is inward-focused: it is a story of politics, treachery and so on; the Others are essentially a sideshow, like Sauron was back when he was still lazing around in Dol Guldur. However, closer reading reveals a picture much more complex than that; fact that things are out of focus does not mean they are not there. In Tolkien, you also have conflicts between good guys: inner Elven politics in the First Age - the conflict between different groups of Elves, the conflict between Sons of Feanor. It is rashness, jealousy and greed of a resident that brings about the fall of Gondolin, not a spy from outside; likewise in the Second Age, it is the pride that causes Fall of Numenor, Sauron merely helped things along.
In the Third Age, there are obviously politics between various good powers, but also within them. Aragorn refuses to return to Minas Tirith as a king precisely because he fears a civil war; and Gondor itself had had its own "Game of Thrones" in Kin-strife, a civil war between Eldacar and Castamir. Much like War of the Kings in Westeros, Kin-strife meant that attention waned towards external threats just as the same were rising up again. Arnor had it even worse, splintering and destroying itself in a series of civil wars between first various claimants for the throne, and then successor kingdoms. But inner conflicts of the Third Age happen in the background; they are there, they are crucial for understanding how things became as they are, but are not the focus of any story. And because most people are only familiar with the primary Lord of the Rings story, and have never read even Appendices, let alone Silmarillion, The Unfinished Tales etc., they are only familiar with primary conflict between people of the West and Sauron. Consequence of this is the (unfair) impression that Lord of the Rings is solely a tale of Good vs Evil told in a very white-and-black worldview. But taken as a whole, opus is much more complex: as Gandalf points out, nothing was evil in the beginning; and Silmarillion proves that not even Morgoth was evil. There are politics, backstabbings, betrayals, stuff that would make Martin proud. Meanwhile, Martin's world is inward-focused: one of the reasons for why the White Walker arc on the TV was so disappointing is the very nature of the story. It is a political drama, a story of plays for power. "Power resides where men believe it resides". The Iron Throne is the Ring of Power, but instead of seeking to destroy it, everybody is seeking to take it, bringing into focus internal political struggles that never were the focus of Tolkien's work. Even in the First Age, the focus was on taking Silmarils back from Morgoth, and any infighting and scheming was incidental, and a distraction from, rather than in service of, the primary goal. Overall, Martin's approach is historological, whereas Tolkien's is mythological. Martin details a lot of things in extreme detail, whereas Tolkien left many things unsaid, unclear or vague in order to create a sense of mystery.
Tolkien and Martin both warn of dangers of concentrated power. Tolkien himself has stated that:
That is something we see in both Lord of the Rings and A Song of Ice and Fire. Aragorn and Stannis are actually quite similar in that both of them view the throne as their duty, not as privilege. Likewise, while Boromir seeks power (One Ring) and has "wanted to be a king", Faramir - consistently portrayed as wiser of the two - outright rejects both the Ring and the idea of him ever being a king, as rightful king is yet to return to the throne (and that is another Aragorn - Stannis parallel, though Aegon and Daenerys also see themselves as rightful rulers).
In both Lord of the Rings and A Song of Ice and Fire, victory is ultimately hollow. As Gandalf points out, evil has many faces and many natures, and whenever one is defeated, another takes its place. It is a never-ending struggle, a concept which ASoIaF embodies well. (It is not very well known that Tolkien had been planning to write a detective story set in cca 100 FA, concerning the Cult of Morgoth; I think the title is "the New Shadow"). Likewise in ASoIaF, victory seems to replace one tyrant with another. Morgoth was followed by Sauron and Ar-Pharazon, and Aerys was followed by Robert and Joffrey.
In another interview Martin has stated that Lord of the Rings involves a lot of "good versus evil" imaginery that had not fared well in hands of less talented authors, and that is definitely true.
Main problem I can see with A Song of Ice and Fire is that it has already set up a Lord of the Rings-style confrontation despite trying to avoid LotR tropes. Of course, I am talking here about the Others. Much like Sauron and his orcs from Lord of the Rings, the Others are bound to push all internal politicking and power games into background. And for a series that has so far relied on those power plays to provide plot, that may not be a good thing at all. It will be interesting to see how Martin solves that particular problem, seeing how it already screwed over TV series.
One particular point, which inspired me to write this to begin with, are parallels between Aegon VI / Aegon Blackfyre(?) and Aragorn. Discussed more here, but both of them had been raised and given education necessary to be ideal kings. What Illyrio says of Aegon - "shaped for rule", "knows that kingship is his duty", and has been living as a commoner for long stretches of time - all apply to Aragorn as well. Aragorn had been educated in Rivendell, but also spent a long time as a "mere" Ranger; Aegon was likewise educated for a ruler, but spent a long time as a commoner. But unlike Aragorn, in all likelihood Aegon is not the rightful king of Westeros; rather he is "stealing the thunder" from Jon and Daenerys. Even so, much like Aragorn has Narsil, so does Aegon have Blackfyre, ancestral sword of Targaryen royal house. Meanwhile, Euron Greyjoy is a Sauron or maybe Saruman parallel: he dabbles in sorcery, leads people who are apart from common run of mankind (Dunlendings / Ironborn) which live segregated from their cousins (Rohirrim / Westerosi), and even has a "dark and a terrible eye". All four characters come into story from essentially nowhere (except no, not really). Both Aragorn and Aegon had been raised as deus ex machinae, set to take the throne when required for salvation of the Middle Earth / Westeros. Meanwhile, Aegon and Euron both invade Westeros when it is ripe for conquest, both plan to marry Daenerys, and are invading the same places. Essentially, it is a mummer's show.
One could say that Martin had attempted to deconstruct Lord of the Rings, but even if that is true, he clearly did not know of or understand its larger themes and universe it is set in. For example, some have stated that Targaryen incest bringing madness is a commentary on Lord of the Rings and importance of bloodlines in there (e.g. Numenorean kings living 400 years compared to 200 for rest of the populace); but closer reading reveals that is not the case. In fact, LotR itself discourages incest - it is pointed out that later Numenorean kings marrying within their own family is a very bad thing - and an argument could be made that Gondor fared better than Arnor because they were more ready to accept local population into fold, even if Numenoreans remained nobles among them.
Also, Sean Bean dies in the first act of ecranizations of both stories. Plus, Tolkien never finished his work (he did finish Hobbit and Lord of the Rings, but those present a minor part of Tolkien's opus); I only hope Martin will finish his own.