Old French and Old Occitan (and modern French and modern Occitan, for that matter) are related, but different languages. They are both descended from Latin.

Languages spoken in northern France are often described as “langues d’oïl”. Modern French is the standard form of the language today but there used to be other languages/dialects as well, such as Picard, Norman, Gallo…those languages are still spoken but the only one that is “official”, the written language and the standard spoken language in schools and the government and on the news, is the dialect spoken around the royal court in Paris. They’re called “languaes d’oil” because of the word for “yes”, “oïl”, which comes from Latin “hoc ille”. In modern standard French it’s pronounced “oui”.

In the south the languages are called “langues d’oc”, also from the word for “yes”, which is from Latin “hoc”. There were lots of medieval Occitan dialects too, but the south of France was gradually incorporated with the north of France starting in the 13th century, and since the 18th century the “standard” French of Paris has been imposed on the rest of the country. So there isn’t really a standardized form of Occitan like there is for Parisian French.

Similarly, in Spain, there is “standard” Spanish, Castilian, i.e. the language spoken by the royal court in Castile. There are other languages spoken in Spain, such as Catalan, which is very similar to Occitan, since they both developed in a similar cultural area along the Mediterranean coast.

Troubadours wrote in Old Occitan because they were from southern France and that was the language they spoke (or, if they were from elsewhere, they chose to write in Occitan for stylistic reasons). There were poets in the north who wrote in Old French though, trouvères (the same word as troubadour, but the way it was pronounced in the north).

Here’s an example of Old Occitan, from William of Tudela’s poem about the Albigensian Crusade:

El nom del Payre e del Filh et del Sant Esperit / Comensa la cansos que maestre W. fit, / Us clercs qui en Navarra fo a Tudela noirit. / Mot es savis e pros, si cum l’estoria dit; / Per clergues e per laycs fo el forment grazit, / Per comtes, per vescomtes amatz e obezit.

(In the name of the Father and the Son and the Holy Spirit / Here begins the song that Master William made / A cleric who was raised in Navarre, in Tudela / He is very wise and noble, as the story says / He was warmly welcomes by clerics and laymen / He is beloved and heard by counts and viscounts)

And for an example of Old French from around the same time, here is a passage from the Conquest of Constantinople by Geoffrey of Villehardouin:

Mult s’acorderent li Venisien que les eschieles fussent drecies es nés et que toz li assaus fust par devers la mer. Li François disoient que il ne se savoient mie si bien aidier sor mer com il savoient, mais quant il aroient lor chevaus et lor armes, il se savroient miels aidier par terre.

(The Venetians were unanimously of the opinion that the scaling-ladders should be set up on the ships and that the whole attack should take place from the sea. The French said that they could not operate nearly so well at sea as they [the Venetians] could, but [that] when they had their horses and their weapons, they would be able to operate better on land.)

They do look kind of similar, right? Some things are the same. Modern French would use "très" or "beaucoup" where Old French used "mult" (or "moult" or "molt"). And the Old Occitan poem has a similar word, "mot". They both come from Latin "multum" (and Italian still uses "molto" today).

There are a few ways to tell that one is Occitan and one is French though, by spelling and vocabulary and sound changes from Latin. For example, Latin “pater”, “father”, becomes “payre” in Occitan. The “t” in Latin disappeared and the vowels surrounding it became a diphthong “ay”. In northern French the “t” also disappeared but the vowels ended up as one vowel, “père”. A very subtle difference, but a difference.

The word “filius” in Latin is “filh” in Occitan. Some strange things happened to the Latin “-iliu” sequence of letters in the various Romance languages. In Italian it’s spelled “figlio”, Spanish “hijo”, Portuguese “filho”, and Occitan “filh”, all indicating that there’s a sort of “ya” sound there. Northern French actually lost that sound in the word for son (“fils”), although it’s still there in the word for daughter (“fille”, even though the spelling doesn’t really show it).

“Canso” in Occitan is the same word as “chanson”, both from “cantio” in Latin. In the south the hard “c” sound didn’t change, while in the north it became “ch” (pronounced like English “ch” at the time, which is actually where our “ch” sound comes from, although in modern French it’s pronounced like English “sh”).

“Amatz”, “beloved”, is from “amatus” in Latin. The vowel at the end dropped away and the surrounding consonants became “tz” in Occitan. In the north, the whole ending changed, into more like an “eh” sound. The first vowel stayed the same in the south but in the north it also became more like man “eh” sound. So “amatus” became “amatz” in Occitan, but “aimé” in French.

The differences can be subtle sometimes! Old Occitan and Old French were a bit more similar than the modern forms. (I probably could have picked better examples that are much more different...)

Hopefully that helps. It's really more of a linguistics question than a history question. Basically, Occitan and French are different languages. They're related but they both evolved separately from Latin. Old Occitan is not a precursor of modern French.

My main source is Peter Rickard, A History of the French Language, 2nd ed. (Routledge, 1989), but it's hard to give a specific source because a lot of this is just stuff I learned in school and afterwards, just from working with texts in medieval languages.