There seems to really be a very minor difference between analytic and isolating languages. A lot of the time I just don't see isolating used at all but analytic used instead. Generally I also see Mandarin Chinese referred to as analytic, but once I saw it called isolating. What's the difference? Are they just synonyms?
Sapir (1921) is the more commonly cited source about this issue, and he sets up two typologies
Analytic/Synthetic/Polysynthetic: These refer to the degree to which different parts of a sentence are fused together to form single words.
Isolating/Affixal(Agglutinative+Fusional)/Symbolic: These refer to the techniques used for encoding grammatical information. From Sapir:
Those languages that always identify the word with the radical element would be set off as an “isolating” group against such as either affix modifying elements (affixing languages) or possess the power to change the significance of the radical element by internal changes (reduplication; vocalic and consonantal change; changes in quantity, stress, and pitch). The latter type might be not inaptly termed “symbolic” languages. ... We may designate the two types of affixing as “fusing” and “juxtaposing.” The juxtaposing technique we may call an “agglutinative” one, if we like.
So French is considered an analytic language because various parts of the sentences are cut up into words, but it is also considered a fusional language (not an isolating language!) because we use fusional affixes (e.g. ils chant-ent, 3pl sing-3spl) to express information. Mandarin is both isolating and analytic.
However, note that in recent years, there is less focus on putting languages into (somewhat arbitrary) macro categories like these, and instead there is more interest in looking at the typology of particular constructions, so one hears quite frequently about the analyticity and syntheticity of particular constructions.