Italian, French, and Spanish (among a couple dozen other languages) began diverging from each other at different places and time periods, with different influences (obviously).
Italian is among the youngest of the Romance (meaning of Rome, not "romantic"), since Latin is native to Italy, it took longer to evolve, experiencing fewer new influences than most of its sister languages.
French is among the oldest, being right next to Italy and one of the first places (Gaul, now modern France) the Romans colonized.
Each area of the Roman empire that ended up producing a new Romance language had different native peoples and occasional conquerors, one reason why the languages developed differently.
Also, pronunciation is among the first things to change as languages evolve (look at the many dialects of English around the world. Many countries have many dialects within them). Also note that all languages develop from the dialects of the common people (not standard forms), often including slang.
All modern Romance languages have simplified greatly from the highly inflected grammar of Latin, but to varying degrees.
Italian and Spanish nouns and adjectives, for example, still consist of a root and a suffix, but they use fewer suffixes than Latin did and use slightly different sets of suffixes.
French, on the other hand, often eliminated suffixes, often leaving just the root, or a shortened or otherwise altered root. There are still suffixes, but there are many more than in Italian and Spanish. They are used less often and many are quite different.
house in Spanish and Italian: casa. (KAH-sah in SP; KAH-zah in IT). - from a Latin slang word for house. (The root for house in standard Latin was dom- )
house in French: maison (from an extinct Germanic language)
There are many other instances where two or all three used either different Latin roots or borrowed words from other languages.
An example where the languages are still similar:
It: amico (ah-MEE-koh). male friend. female: amica. plural of amico: amici (ah-MEE-tchee); plural of amica: amiche (ah-MEE-keh)
Sp. amigo (ah-MEE-goh), male. female: amiga. Just add -s to form the plurals, pronounced as a hard S (as in sip).
Fr. ami (ah-mee). male. female: amie (e is silent. still: ah-mee). Adds a silent -s to form the plurals.
Note that French pronunciation is quite distinct from the other two, because it is mostly syllable-timed (most syllables get equal stress), whereas the other two are mostly stress-timed (usually there is a stressed syllable in a word).
French also, far more than the other two, often requires that sounds be added, removed, changed, and/or moved beyond syllable/word boundaries.
Fr: les amis (the friends): pronounced as leh-zah-mee
It. gli amici (roughly: lyi-ah-MEE-tchee)
Sp. los amigos (lohss-ah-MEE-gohss)
It's also interesting to note that classical Latin had no words for THE, but all its daughter languages do (more than one word).
The same thing will happen to English, eventually. Either hundreds (most likely) or thousands of years from now. English as we know it today will be dead, but it will have at least a few if not many dozens of new Anglic daughter languages: mutually incomprehensible to each other. Some of them will undoubtedly sound quite different from some of the others.
Note also: there are many grammatical differences (as well as many similarities). For example, all three languages have different numbers of tense/mood/aspect combinations. French has fewer than the other two, and far fewer inflected forms, comparatively speaking.
studied linguistics and the history of languages; taught French; intermediate Italian; frequent exposure to Spanish (know enough to be dangerous).