Both seem to be words for a well or a shaft. Google Images shows similar pictures for both words. So I am confused if there really is any difference between them at all. Is it just a matter of regional preferences? What is the more commonly used word in Latin America (Mexico in particular)?
An aljibe is an underground tank in which water is stored. It is usually man-made and was mostly used to collect rainwater in the past or in areas without running water. In cities there were big aljibes filled up with water from nearby rivers, which was useful for times of drought and sieges. Nowadays some country houses have aljibes that are refilled with water from water tankers. There are a few more less common meanings for aljibe: http://dle.rae.es/aljibe
A pozo, when talking about water, is usually a (man-made) hole dug in the ground in order to look for water, so pozos are not a form of storage, but rather sources of water. It can also just be a deep hole in the ground (a pit). Pozo has many more specific meanings: http://dle.rae.es/pozo
The previous answer is not correct for proper Spanish (e.g., Castilian Spanish from Spain).
Regardless of the regional modes used in some Latin American countries, especially in Central and North America, e.g. Mexico, aljibe is the correct word to describe what in English is a bored water well.
A pozo is a generic word that simply means 'hole in the ground', so it does not necessarily refer to a well unless it is accompanied by other words as in 'pozo de agua', which literally means 'hole of water' (which clearly refers to a water well).
But to call a well 'pozo' is technically incorrect; it is just an idiom, or deformation of the Spanish language.
For example, when a grave is dug at a cemetery, the workers are actually digging a 'pozo' (that is, a hole in the ground) that will be used for a grave (it is not a grave until a body is put in it).
Similarly, when someone is going to plant a tree, she/he has to make a 'pozo' in the ground to fit the roots of the tree. Conclusion: the only correct Spanish word to translate the English words 'water well' is aljibe.
(PS: I am a native Spanish speaker with a masters degree involving linguistics from Universitat Autonoma of Barcelona).