When a molecule has plane of symmetry, or centre of symmetry it is termed as an achiral compound (though it contain some chiral centres). Same goes for the meso compounds due to the internal compensation. So both terms mean exactly same. But I am eager to to know the applications of these terms in different cases. All meso compounds can be told achiral and as well as all chiral compounds as meso?
From the Gold book:
A term for the achiral member(s) of a set of diastereoisomers which also includes one or more chiral members
A compound that can be superimposed on its mirror image.
The difference lies in the presence of the chiral centre. All meso compounds are achiral (due to internal compensation as you have mentioned) but all achiral compunds aren't meso.
Look carefully and you'll be able to spot chiral centres in all the above compounds but you'll also notice that they have a plane of symmetry (internal compensation) which makes them achiral.
Bromocylohexane has a plane of symmetry which makes it achiral. But it has no chiral centre, that's why it doesn't qualify to be a meso compound.
A meso compound is one which satisfies both of the following criteria: (1) it must have chiral centers and (2) it must have an internal plane of symmetry. On the other hand, an achiral compound is one which has an internal plane of symmetry. Thus you can say because a meso compound has an internal plane of symmetry, it is in fact achiral and optically inactive. You could not however say that all achiral compounds are meso... the terms are not synonymous. For an example, benzene is achiral but not meso. It has many internal planes of symmetry but does not have any chiral centers. It does not meet the first criterion for being meso. All chiral compounds cannot be considered meso as you suggested. The presence of chiral centers does not necessarily mean a compound is chiral.