As an owner of both, the actual game of AGOT is a bit better. They are both good games, but different.
Both are character/item/location/event centered strategy card games. AGOT is more managing your money, lands, and minions over the course of a round, while getting attacked either through Military force, Intriguing actions, or Stealing Power. LOTR can be boiled down to a complex round of blackjack, (though you'd never know it), while avoiding and mitigating impending doom.
You "make progress" or "gain power" in both these games to a mark how far you've come and how close you are to victory.
LOTR Official solitaire is alright, but the game's scenario is limited to what you've bought, whereas AGOT is one deck vs one deck, and many decks can be made from one's card pool. Multiplayer is essentially necessary for success in LOTR, and is a fun change of pace in AGOT.
Even if you are going to be okay with just the scenarios that come with the LOTR core, you'd have to buy expansions to have enough different cards avaliable to have a shot at surviving those scenarios with any consistency. Many people play solitaire "double handed" to allow more card synergy.
Challenging is good, but don't think you are "getting a better deal" by choosing one or the other in terms of how much money spent. They will both require some development (buying expansions) to get to even the lower levels of potential when building decks, not to mention keep things interesting. I own maybe 40% of the total AGOT card pool and have several great decks with plenty of room for interchangeable themes. The 2nd edition card pool is obviously unreleased as of yet.
Unless you live somewhere with tournaments and plan to actively participate in them, ignore those saying AGOT Is broken or needed a redo. Issues of that type will only arise if and when you are playing with and against others using a complete card pool. Not the case with the vast majority of players.
The 2E will likely not be compatible with 1E, and despite being a better and more in depth game than LOTR, that is a huge turn off for AGOT and a big selling point for LOTR.
Anyway, I've found these games to be high quality. Hopefully you do too.