Like when you watch a character talk in anime you'll notice he/she is not lip syncing the words he/she pronounces: their mouths move up and down and appear to be looping. This limited animation technique is 'looping' from talking, running, walking, etc. So why is it Japanese animation is limited?
Please see the latter part of the answer to Is intentionally producing a lower quality TV release unique to anime?:
American cartoon viewers are very particular about the quality. My friend who is a key-frames animator for The Simpsons told me that there is one animator whose entire job is just to animate the mouth movements, because if the mouth movement does not perfectly match the voice actor's words, the fans generally feel that they cannot suspend disbelief. In contrast, anime in Japanese does not always bother to record the voice actors' voices first before animating, since Japanese fans do not care much whether the mouth movements exactly match. . . . American viewers are used to watching mouth movements like in The Simpsons and Disney films which are perfectly aligned.
This is not a case of anime being "limited" in a negative sense of slacking off, cutting corners, or being too rushed, but rather, Japanese viewers are not at all bothered by mouth movements which do not perfectly align with the voice, and therefore do not expect that out of animation production companies. It is a cultural difference, so it does not make sense to label the Western style as "better at" animating.
Anime can precisely match mouth movements to the voice when it is desired to do so (see the opening theme song sequences of Ao Haru Ride and Ore Monogatari!! for examples), so the companies have the required skill to do it. It is just a matter of preference that they often do not.
Because Japanese viewers are not particular on this point, it allows the production company to begin animating the cels before the seiyuu (voice actors) finish recording their lines. This gives the anime industry more flexibility in the timeline of production, as compared to non-Japanese animation which requires the script to be finalized and the voice acting to be recorded before the mouth movement animation can begin (in other words, multiple aspects of the production of an anime episode can occur at the same time). A benefit of this flexibility is that seiyuu have the freedom to ad-lib lines in the studio on the spot if they come up with something they think might work better than what is written in the script that was handed to them; even if the mouth movements have already been animated to basically match the line in the script, improvisation is still possible. Some fans enjoy buying anime scripts and comparing the original script to the final product, so they can discover which parts the seiyuu contributed.
Historically, anime was drawn by hand, so looping, panning, overlaying cels, etc. were more utilized in older titles (since animating a whole episode's worth of cels took a lot of man hours) than in current anime, which is predominantly computer-animated. If you watch older series such as Ashita no Joe or Ace wo Nerae!, you can see very impressive and creative alternatives to animating many frames per second, an art form which has largely diminished with the advent of computers. There is a remarkable and arresting charm to some of the classic animating techniques which were developed in order to stylistically or realistically convey the scene or the character's emotion in less frames. The looping mouth movement is one of these early innovations.
Edit: I have done a big re-work on this answer, so it is worth re-reading if you have read an older version. This answer also focuses on mouth movements in particular.
The more time that is spent dubbing a show precisely, the more money animators will need to be paid, as they will be drawing many additional frames. Additionally, retakes are harder for voice actors/actresses - so more money may be needed for the time needed to re-voice a part as it is more difficult to match the precise mouth movements. Anime usually have tight budgets, so sometimes paying for more animation or voice time isn't seen as a worthwhile decision.
There are a lot of techniques that studios use to save money, and this is just another one of them. Other examples include using the same drawings in different scenes, not showing lips at all when characters are talking, panning shots (over one drawing).
Even in early anime, looping was common as a cost-saving technique. (Kimba the white lion)
Animating people talking in a fixed loop is a fairly standard technique in animation. There are a certain number of mouth positions to know, which can also be simplified down to as low as two looped frames. The broadness of these simplified frames lets the timing and positioning seems less off than a highly precise matchup which may seem unnatural. Sometimes in a close-up shot, there will be a larger range of mouth poses that the animators will draw to add detail to the scene. In most scenarios though, animators can get away with a reduced set of movements
Differing levels of mouth movement:
A small guide to animating dialogue. This isn't a feature of just anime, but is in most animated and 3D works too.
In regards to voice acting itself, the way the English and Japanese voice actors/actresses work are quite different;
Usually when the anime is first being made, the Japanese actors will face a Leica Reel (sometimes called animatics in the West, but essentially timed storyboards). There are sections in the script where they can ad lib and generally, the animators will match the pauses and openings with the recorded voices.
In the exported market however, the anime is already created and voice actors have to "match the lip flaps" that already exist. The translation team translates the script - but usually the non-scripted ad lib sections will be translated too. This is because a lot of the time Western voice actors won't have much of an idea about the story (at least in the less prolific studios / actors).
Western dubbed anime can sometimes seem better dubbed (in term of lip matchup) because of the sole focus on this aspect of the show. Japanese studios need to worry about a lot more aspects of the anime and cannot afford to be perfectionists in each area. Seemingly (I have no solid source for this), culturally the Japanese experience less of a disconnect from the show with less-than-perfect syncing than in English.