Omission of definite article with musical instruments
Solution 1:
They're pretty much equivalent.
That said, omitting the article has a slight feeling of playing with a group or orchestra, wherein the instrument is a synecdoche referring to the position the person occupied within the group.
I used to play the flute.
I used to play flute in the Civic Orchestra.
Omitting the article also can carry the feeling of playing an instrument in the general sense.
I play woodwinds.
In this case it would sound strange to use the article because you are speaking of a class of instruments.
Solution 2:
I remember when I first moved from the UK to the US, I was quite taken aback by the “I play piano” usage, without the article. So I’m pretty sure that this usage is very uncommon in the UK (at least among classical music circles). In US usage, Robusto’s answer, that the article-less usage is more common in reference to playing with a particular group, fits my experience (classical groups, north-east US) pretty well.
Solution 3:
Articles are creatures more of usage and discourse than of grammar. The human speech communities involved (UK vs US; musicians, cooks, scientists, etc.) have certain patterns and expectations for use in the domains they control. The surrounding text (conversation or writing) also guides usage.
As an American, I accept using or omitting the article before an instrument; they are nearly interchangeable for me. I like the idea above that the article-less usage stresses playing with a group and the article usage stresses the position within the group. As an ESL teacher, I have generally taught that omitting the article highlights the action or activity, almost as if practicing-violin were a single intransitive concept, like swimming. Using the article gives a subtle shift in focus to the instrument. The following is a sentence I would be likely to produce; I would accept any version of this (article-wise) that I might hear:
"When he was little he played violin, but he switched to the cello when he got to middle school."
Solution 4:
I heard somewhere that if the instrument is big and can hardly or even can't be carried (like cello, piano, drum set, harp, grand-piano, organ, etc.) the definite article is needed. I am not sure if that's correct, though.
Solution 5:
I actually play the mandolin in a group and it is a small light instrument that can be very easily carried in a small case, more easily than a guitar so this argument is not good enough. I agree that "I play mandolin" seems to be a modern way of saying "I play the part devoted to the mandolin in this group". "I play the mandolin" is the best way to specify which instrument I play and the normal classical English usage.