I find TtA to be the better game, both in terms of fun and ease of play. The reason for both is the theme which--unlike TM which is a convoluted mess that really seems to exist only to hang mechanisms on--is a Civ building game. If you've played any empire or civ builder, TtA will be easy to pick up. Winning is another story, but everything in TtA "makes sense". For example, you're short on ore, which is what you use to build new things, it's easy to realize that you need to build more mines or, better yet, upgrade your mines. There's none of this, "but this turn I score points like this and to do that I need this to turn into this, which becomes this and then I can build this" that you get in TM.
You instinctively know that Wonders are good, even if you haven't figured out the best strategy for building each of them. Leaders, too. You know that Rockets are more powerful than Cannons, or that Riflemen are more powerful than Spearmen. Computer labs produce more science than an alchemist would, and that having an Air Force is going to really bump up your military if/when you build one. So, I find TtA to be easier to teach with the only real complication coming in the production/corruption phase. Luckily the new version of TtA (and the app) simplify this dramatically, and it's fairly simple to see if you'll face corruption or hunger in your civilization with the option to redo your turn and see if you can do things different to mitigate it.
Strategy is an entirely different beast, as @Hardco will attest I haven't figured it out in well over 150 games played. I don't care, because even when losing you're building a civ and the games have that "epic" feel you expect from PC titles like Civ. I might have lost, but look what I built! YMMV on that, but that's why I keep playing even though I'm getting my ass handed to me in every game.