This is a similar difference to that of saying アメリカ (Modern sound-conversion of a word) and [米国]{べいこく} (Using kanji that previously represented sound-conversion of a word through ateji).
[仏語]{ふつご} is actually a shortened form of [仏蘭西]{ふらんす}語{ご}, with 仏蘭西{ふらんす} being archaic [当]{あ}て[字]{じ} (kanji characters used in place of kana) for now-katakanized フランス.
The [仏]{ふ} in [仏蘭西]{ふらんす} just so happens to have an [音]{おん}[読]{よ}み (Pronunciations derived from Chinese origins) of [仏]{ふつ}, giving us [仏語]{ふつご} as a shorthand.
Therefore, it is safe to assume that [仏語]{ふつご}, due to it having an older origin, is technically considered more "formal" speech than フランス語, similar to how [米国]{べいこく} is used in news while アメリカ is rarely mentioned.
As a general rule of thumb as well, kanji is considered more formal than using kana.
Hope this helped!