These two terms appear to be used nearly interchangeably to refer to languages with little morphology. However at the same time I sometimes see them contrasted with each other, but cannot identify a pattern.
Isolating languages do not use inflectional morphology, i.e., using affixes or other manipulations of roots to create words--all words are seperate, and none contain multiple morphemes. Vietnamese is an isolating language.
Here's an example: the word snowman. It has two morphemes: snow- and -man, and therefore has a morpheme-to-word ratio of 2:1. Or take Spanish corremos, we run, which has three corr-, -e-, and -mos, for the word, present tense, and person/number. (Please correct me if I am wrong--I don't know Spanish), so its ratio is 3:1. In an isolating language, this ratio approaches 1:1--words stand alone and cannot be separated (hence "isolating.") So, these concepts--"snowman" and "we are running"--would take multiple words to express, as each one would contain only a single morpheme. Isolating languages have no inflectional morphology and no derivational morphology.
However, analytical languages are a broader category; they have a low morpheme-to-word ratio as well, but--and here's the big but--they can have a derivational morphology. This means that while "we are running" would still be multiple single-morpheme words like in an isolating language, the word "snowman" could be a two-morpheme word.
Thus, isolating languages are actually a subset of analytic languages. Analytic languages cover languages with no inflectional morphology and a low morpheme-to-word ratio, while isolating languages have all that and no derivational morphology.
When I've seen a clear distinction made here, it has been the following:
- Isolating languages have a very low morpheme-per-word ratio
- Analytic languages have little inflection
However, this distinction is not made very consistently, so they're often used interchangeably. I have also seen a different distinction made, where analytic languages have a low morpheme-per-word ratio and isolating languages have a morpheme-per-word ratio of 1:1, though this one is less common in my experience. The fact that even with this distinction isolating languages end up being a proper subset of analytic languages only makes this problem worse. This has its origins in the definition of "synthetic", the category both of these are typically contrasted with, since it's used both as a description of languages with a high morpheme-per-word ratio and as a description of the superset of terms like fusional and agglutinative, both of which describe languages with lots of inflection.
In practice, conlangers seem to make too big of a deal out of these sorts of labels. They're there to be descriptive, so it's not a problem to discard them when they aren't, and they're often far less informative than most who use them think.