More than I care to list. Basically, all of Western Europe and it's relevant colonies have latin roots, exceptions being Germany, The Netherlands, and England. English has latin roots, but mostly just by way of the Norman invasion in 1066, not due to Roman influence. Mostly.
Irish Gaelic, Welsh, and other British Isles languages are completely unrelated to Latin.
Of course, the most widely spoken language across the globe outside of Mandarin Chinese is Spanish, and it of course has heavy latin influence.
As for why we don't all just start speaking latin; Latin is very much like french. Lots of rules, with lots of exceptions to those rules. Lots of history between the last time anyone spoke latin. The fact that the latin you read in latin text books or historical texts is not a kind of latin that anyone really spoke is also an important point.
http://en.wikipedia.org/wiki/Classical_Latin This is what they teach in High Schools and Colleges.
http://en.wikipedia.org/wiki/Vulgar_Latin This is what most Latin speakers in Antiquity actually spoke, and what eventually transformed into modern Italian.
In addition, looking at the spread of regional dialects across Italy shows that even Modern Italian isn't really what is spoken everywhere in Italy. We in America have it easy, there's really only two languages worth knowing to communicate effectively; Spanish and English. This is because there was never really a point in United States History where populations lived in any real kind of isolation, without any communication beyond their region. We've only had 234 years, after all, and for most of that time, we've had means of rapid transportation.
In Europe, the situation is far more complex. Language across a single nation can vary wildly, and people need to learn many different languages just to get through life without looking like an idiot.
http://upload.wikimedia.org/wikipedia/commons/0/06/Italy_-_Forms_of_Dialect.jpghttp://en.wikipedia.org/wiki/File:Langues_de_la_France1.gifhttp://upload.wikimedia.org/wikipedia/commons/5/5b/Continental_West_Germanic_languages.pngThe above just manages to scratch the surface. All of these have deep historical roots, and no one wants to give up something they grew up with.
Latin is not an ideal language anyway. The closest to an ideal that anyone has created is probably Esperanto, just due to it's simplicity and easy to learn linguistic rules with no exceptions.