I don't know if I'm a Tolkien expert, but as far as I know, the fell beasts were basically pterodactyls; I think unintentionally by Tolkien at first, but he later leaned into it. They were awful and terrible, but they were also very much natural creatures.

The great eagles, on the other hand, were potentially Maiar--the same kind of pseudo-divine, pseudo-angelic being that the Wizards are (and Sauron and the Balrog, notably). If I recall correctly, Tolkien went back and forth on whether or not they actually were Maiar, but even if they weren't, they were at least elevated by the Valar.

If the eagles were Maiar, they would be nearly unkillable by normal people in Third-Age Middle-Earth. As far as I can remember, we only see the deaths of four Maiar during the Third Age, and all in the last few years of it: the balrog known as Durin's Bane (killed by Gandalf), Gandalf himself (killed by the balrog), Saruman (killed by Grima, but only after having his powers stripped by Gandalf), and Sauron (killed by his own hubris by putting all of his power into the One Ring). It seems like it takes a Maia to kill a Maia, at least in the Third Age.

The fell beasts, on the other hand, are very killable. Legolas casually takes down one of them with an arrow (one!) after Lothlorien, and Eowyn beheads one almost incidentally on her way to the Witch-King. The rest are killed in the eruption of Mt. Doom after the destruction of the Ring, which the eagles survived; both of which are kind of immaterial to the point, the fell beasts being right over the top of a magical volcano while the eagles were half a Mordor away when it happened.

But it does bring us back to the Eagles; because while the Lord of Eagles does mention "risking ourselves" against the "great bows of yew" wielded by the men around Mirkwood during the time of The Hobbit, it seems like he was being a bit over-cautious. Even if the Eagles weren't Maiar and were technically killable, we see the Great Eagles participate in (by my count) three battles: the rescue of the Dwarves from the Goblins and Wargs outside Goblin-Town, the Battle of Five Armies outside Erebor, and the Battle of the Black Gate at the end of Return of the King, but we don't see a single Eagle killed or even wounded in any of those affairs.

But the eagles were also a lot bigger.

Tolkien was vague about the fell beasts' size, but we can make some assumptions; it had to have been small enough for Eowyn to cut completely through its neck in a single stroke, and leaving aside the question of her strength, the neck certainly couldn't have been thicker than the sword was long. Assuming that the Rohirrim used swords like the early medieval Anglo-Saxons they're based on, that means a blade length (and thus a neck thickness) of about 75cm/2.5 feet at most.

But there's no chance Eowyn could use the entire length of the blade. That's not a statement on her gender, nobody could do that. In the real world, the maximum would be something like 15cm/6in, but this is a heightened fantasy world. Let's say Eowyn--empowered by her love for her uncle and the power of the very Valar themselves--could use more than half of that (very sharp!) blade to chop through a credulity-straining 40cm fell beast neck. To do some really sloppy math, a cobra chicken Canada Goose has a neck thickness in the ballpark of 5cm, which means that we can easily multiply its dimensions by about 8 to see what a reasonable size for a fell beast might be: the goose's wingspan of 125cm then becomes a 1000cm/33ft fell beast wingspan. Certainly formidable for the real world; that would make it about three times the size of the largest living flying bird, the Andean Condor, and about 1.5 times the size of the largest flying bird ever, Pelagornis miocaenus.

But the eagles were significantly bigger. Thorondor, the mightiest Great Eagle of the First Age, was said to have a wingspan of 30 fathoms, which is an insane 55m/180ft. Let's assume the Great Eagles got significantly smaller by the end of the Third Age; they'd still need to be big enough to carry two emaciated hobbits, or one full-grown wizard, or one very fat dwarf. (I'm assuming here that the ringwraiths, being wraiths, had no meaningful mass and thus we don't have to apply the same calculation to the fell beasts.) If we do a lot of handwaving with the average size of a normal eagle and its maximum load, and try to scale that up to being able to carry Bombur, and just sort of vaguely nod at the square-cube law, we end up with Gwaihir needing a wingspan of about 1500cm/50ft, significantly bigger than the fell beast!

Couple that with their undefeated battle record, including one where they fought against the fell beasts, and the Great Eagles are clearly more formidable.

(sources: 1,2,3,4,5,6,7,8, and probably more I don't remember. Side note, look at this hilarious early drawing of a pterosaur. It looks like a winged possum.)