It belongs to the "synthetic" language group, which means that unlike English and other "analytical" languages, different grammatical aspects are expressed in one word by changing the structure of that word - adding an ending or prefix, modifying the core of the word, etc. In analytical languages such as English, the same is achieved by using separate auxiliary verbs, pronouns or adjectives while the actual word remains unchanged. In Czech, one word is often sufficient to express what English can only achieve by using multiple words. Source
I see the difference that j3st3r is making, but maybe the reason they're interchangeable (see here as well) is that it seems that the more analytic a language gets, the more isolating it'll become. English seems to be a relatively analytic language, especially compared to heavily inflected ones. It uses separate words like "will" for the future tense, prepositions like "of" and "to" instead of cases (even the possessive is particle-like, since you can say things like "the Queen of England's crown" which shows it doesn't act like a case). I wonder if you could have a language that is relatively analytic but not relatively isolating (or vice versa), and maybe that's why the two terms have become synonymous (in English).
I was under the impression that they were archaic because the terms were invented as a way to "describe" languages in the early twentieth century, but in fact the dichotomy of analytical/synthetic/isolate doesn't really tell you much. As modus. irrealis points out, there really isn't much difference between analytical and isolate. As I recall, there was another term - "agglutinated" - to describe word formation common to Ural Altaic languages as opposed to Indo-European. The closer you look the less the categories seem to make much sense.
Probably a more useful description in terms of similarity of structure would be Greenberg's division into "SVO" versus "VSO" versus "SOV" languages.
Indo-European languages are typically synthetic. This means that different shades of meaning are given to words by adding (or subtracting) prefixes or suffixes to them, for example when you decline a noun, or conjugate a verb. These prefixes or suffixes do not exist as independent words; they appear always as part of a complete word.
In analytic languages, by contrast, new shades of meaning are obtained by adding independent particles, which can also be used on their own. Sino-Tibetan languages, including Chinese, are analytical. In Chinese, verbs do not change to show tense/aspect/etc., and nouns are not inflected to show case, gender, or number. Rather, this inflormation is conveyed through other words placed beside them, or through word order (which tends to be stricter than in synthetic languages).
I don't agree very much with the statement that English is analytic, though. I think some scholars are too eager to find originality in English. It has lost most of its older inflections, and is considerable less synthetic than other IE languages, but it still has some synthetic features, such as inflecting nouns to show the plural and the possessive case, or inflecting verbs to show tense and aspect.