Conjugation forms «движется» and «двигается» (both derived from the common infinitive «двигаться») can differ between themselves in 2 ways:
WAY No.1
«Движется» is a unidirectional verb form showing a motion in progress from one point to another, e.g. «(Сейчас) автомобиль движется в сторону центра города» (“A car is moving towards the city centre” meaning “now”).
«Двигается» is a multidirectional verb form showing motions in different directions at different times and in different manners, e.g. «Обычно его трудно заметить, т.к. он двигается очень быстро» (“Normally it can’t be seen, since it moves very fast” meaning a regular action in a usual manner).
WAY No.2
This way shows their difference in meaning. Both forms «движется» and «двигается» may refer to an action in progress happening now, but with the following difference:
1. «Движется», as we already know, shows a motion in progress from one point to another, i.e. a motion in one direction (see the example above); while
2. «Двигается» shows an action, being done by somebody or something, in the sense that a person or object does not have a fixed position, i.e. somebody or some one is flexible, able to move, for example: «Смотри, у него двигается голова» (“Look, it can move its head” with the action being demonstrated now – continuous aspect). However, it may show a regular action, too. For example, the sentence «У него двигается голова» can also refer to a permanent characteristic feature (simple aspect) of smth/smb. In any case, the 2nd form «двигается» is opposed to the 1st form «движется» semantically.