This is sure to be a frequent question here, and, as I've played both, I figured I could offer my own perspective. Before I begin, let me be clear: preferring one over the other is mostly a matter of taste; the two mods simply have very different design philosophies.
So in terms of engine differences, Third Age of course has higher raw graphical quality, but suffers from M2TW's unresponsive troops. In my opinion, Fourth Age has a slightly more tightly focused graphical theme; some of the graphics in Third Age (say, the Osgiliath Pikemen of Gondor) just don't seem to match everything else. I think everyone can agree that the RTW engine has much crisper and more responsive battlemap controls and troop response.
Thematically, both are set in Middle Earth, but that's where the similarities end. Third Age takes its visual cues from the Peter Jackson movies, and the Third Age isn't the most politically complex - it's good (Elves, Dwarves, and Men of the West) vs evil (Orcs, Saruman, and wicked men of the south and east). That simplicity ends up playing out on the campaign map - a largely good West of the map vs a largely evil east of the map. Fourth Age is more confused; there are really only a few reliable alliances, and there are (I think) 18 factions as opposed to Third Age's 12. It makes for a campaign game that tends to play out with more variety, and it gives complexity to regions of the map that are unused in Third Age. For example, in Third Age, Harad is the one southern faction, and it exists only to fight Gondor. In Fourth Age, the Empire of Harad is powerful - but it's hemmed in by Harondor to the north, Far Harad to the southeast, and the nomads of Khand to the northeast. So in Third Age, the Haradrim really only ever fight Gondor, occasionally getting in border scraps with Rhun. In Fourth Age, there are multi sided wars and diplomatic maneuvering in the south, not even counting the Reunited Kingdom's involvement. So Third Age offers a more immediate head on fight - you don't decide who you fight, just how you fight them. In Fourth Age, it's both - which is appealing for players who want to put a lot of thought into campaign map strategy, but less appealing if you just want to get into the fight.
Fourth Age is also much more heavily lore focused. This goes beyond just visuals (straight swords for elves) and flavor text. It's built into every aspect of the game. In Third Age, Elves can raise armies and replace losses without too much more difficulty than men. In Fourth Age, population loss is one of the greatest difficulties facing the Elves. On the battlemap, Fourth Age's elves make Third Age's elves look trashy and weak; there is a lot more contrast between elves/dwarves vs MotW vs Men of Darkness than in Third Age, where that contrast exists, but is considerably more muted. For example, Third Age's Uruk Swords, wearing unimpressive looking Orcish armor, are only slightly worse statistically than Gondor's swordsmen. In Fourth Age, Uruks are better than standard Orcs, but still, as in the books, far inferior to the well armed and trained men of Gondor.
Third Age is visually spectacular, well balanced, and easy to dive into right away. It's designed for players to be able to recreate the epic battles in the LotR movies, and it does a great job at that.
Fourth Age is also well balanced, and has a tougher learning curve that pays off in its greater complexity. It has more factions, easier battlemap control courtesy of the RTW engine, and far greater attention to lore.
I suppose a really (over)simple way to put it would be to say that Third Age appeals more to movie fans, and Fourth Age appeals more to book fans. I'm decidedly one of the latter and prefer Fourth Age, but if someone asked me which one they should play I would simply tell them to play both and make up their own mind.