Professional and amateur violinists, at least in America, not infrequently refer to their instruments as "fiddles". It's a way of down-playing the stuffy image some people attach to classical violin music.
However, if you want to get really technical, there can be a difference in the way that fiddles and violins are set up. Fiddles often have lower, flatter bridges (the wooden piece supporting the strings, placed between the two s-shaped holes in the top of the instrument). That shape makes it easier to play double-stops (two strings at once) and easier to play lower notes. With a "violin", you need a higher bridge to help with playing higher up on the neck, and the higher arch helps keep notes separate and clean - making double-stops less easy, but not that far a reach.
Firstly, "to play the violin" is a standard British way of saying it, whereas you frequently hear "to play violin" in the US. You can check the opinions of some native speakers e.g. here: http://forum.wordreference.com/threads/play-the-piano-play-piano.1564241/
Secondly, there is nothing grammatically wrong with "to play a violin". This expression would merely refer to some violin - e.g., an actual thing rather than the type of a musical instrument. For obvious reasons such usage is much less common.
If you want to have some fun with it, check the title of this book (seemingly written by a native speaker): http://www.barnesandnoble.com/w/learn-to-play-a-piano-robert-masterson/1112100183
It does not. The standard variations are:
- to play violin (US)
- to play the violin (GB)
In the latter case the violin is treated the same way the animal species are treated in English: the lion, the tiger etc. - it's the name of the species, not an individual animal: