Different anime and manga, and really all works of entertainment, have different ways of captivating their audiences. Some of them create mental mementos so strong that they last forever whether you want them to or not, and others leave light footprints that disappear with the first snowfall, but are no less beautiful.
A takeaway from a series is some effect one gets from it that lasts long, long after the last time an episode of the show is actually watched. It’s the kind of thing that can provide you with hours of discussion and can even change your life. For example, take the terrific hypothetical political situation from Legend of the Galactic Heroes (corrupt democracy vs. benevolent dictatorship). That one’s provides some serious food for thought, and I’ve had one political discussion centered around that show that lasted for several hours. Akumetsu and National Quiz provide some excellent manga examples of political takeaways. It’s not limited to higher political and philosophical questions, though. The fundamental takeaway from Hajime no Ippo is a very simple one; “hard work pays off”, but at the same time resonates profoundly (in part due to stellar presentation) and really sticks with most of the people who have seen it.
A moment, on the other hand, is something that one can only reach when actually watching or reading the work. It’s something usually provided by elements other than the screenplay that captivates the one experiencing it. This is, in my view, why people rewatch things over and over again. It’s the essence of why the Rabbit Nabokav bit from 20th Century Boys is still awesome every time I read it; I may know what’s going to happen, but it’s not the twists that keep me coming back. Rather, it’s the booming-thunder intensity of the scene, backed by some of Urasawa’s finest camerawork, that makes me want to read it over and over again. It’s one of the reasons I keep coming back to stuff like the manga of Yoshida Motoi and the To Heart anime; they don’t carry profound or strong messages, but they’re just so beautifully done that simply being exposed to them is sublimely refreshing.
The two are not mutually exclusive. Indeed, series that pull off both, like Honey and Clover or Sasameki Koto or Bakemonogatari, are reasonably common and usually very, very good. The reason I bring this up and make a point of separating the two is that I feel that, while Takeaways are something most people properly value, I’ve seen a lot of people be dismissive of a works like One Outs and Redline, which get accused of providing momentary thrill and nothing else. My counterargument to that stance is that that momentary thrill is a) as hard to produce as a Takeaway, b) easy to regain via rewatching, and c) still a definite impact produced on the viewer by the work. If a series aims to provide a string of moments and does so at a level near the peak of its medium, I, at least, feel it deserves to be praised accordingly.