When it comes to Latin, 'æ' is the same as 'ae', at least when in the diphthong.
When the vowels are in different syllables, as in aer, then 'æ' is not used.
You could see this so that 'ae' is such a commonly appearing combination that it is essentially a single letter.
It makes thus sense to write the two letters as one, and it also has the benefit of indicating that it is indeed a diphthong.
From a Latin point of view 'æ' is typically understood as a ligature, not a letter, but that is a mere matter of definition.
In some languages (at least many Nordic ones) 'æ' is a separate letter and is not interchangeable with 'ae'.
In Latin you can always write 'æ' out as 'ae', and thus indeed Curriculum vitae equals Curriculum vitæ.
The ligature should be applied consistently: either use it in all possible instances or never.
In the texts I have seen it is more common to never use ligatures, and non-diphthongs can be indicated by dieresis as in aër.
The same applies to the ligature 'œ' for 'oe' in Latin, but its use in other languages is different from 'æ'.