Ever since I learned that you could learn to play instruments through a classical or contemporary method, and that musical teachers are taught classically or contemporarily, I've been asking myself the question, "What's the difference?" What's the difference between the two methods? Is it simply a matter of the music being taught or does it have a deeper meaning? Is it in the way you play? Or how you feel when playing? What is the difference, really?
I would say personally that it's mostly a matter of the music being taught. The main exception to that is voice; all you have to do is listen to, say, Pavarotti and Frank Sinatra to tell the difference. They are obviously each going after a very different sound. Operatic singing doesn't use a microphone and emphasizes natural volume. The sound is stylized with a good deal of vibrato as well.
Interestingly, if you listen to Caruso's 100-year-old recordings, you will notice that his use of vibrato is not as consistent as with more modern operatic singers. He usually used it mostly when hitting high and loud notes. This suggests the possibility that the difference between popular and operatic styles became more divergent during the 20th century.
As for other instruments, it is also a matter of going for a different type of sound. "Classical" piano spans 300 years and has a wide range of sounds and techniques. Just to take one example, Mozart's music makes much less use of the sustain pedal than Debussy's. So does pretty much all of Jazz, for that matter. Listen to Art Tatum, pay attention to the left hand, and you'll see that there's very little pedaling. You'll also perhaps notice that all of his fast runs have a sort of detached, non legato sound, meaning each tiny note has an even tinier space of silence between it and the next note.
The reason that popular musicians often mention that they are "classically trained" is to convey that they have "chops", meaning that they have a solid grounding in technique and theory. It's hard to improvise your way very far in classical music; it's sort of like when you were a kid and sat around pretending to read, and your older sister called you out and said you were just looking at the pictures and making stuff up. (Well, that happened to me when I was a kid, anyway.)
I think the 'classical' approach is much more 'proper', as in in dots the t-s and crosses the i-s. Attention to the written detail is very important. Take any music played in a 'classical' exam. Every note must be given its exact timing, dynamics need to be followed, etc. Whilst there certainly is nothing wrong with this concept, it makes each performance of a particular piece very similar.
The 'contemporary' approach is to personalise each piece more. As in, use the music as a guideline rather than a blueprint. One is allowed to interpret what the composer has written in one's own manner, style, etc. Rather than saying ' That's what he wanted', it's more of a 'Maybe he didn't think to do it like this. Let's try.'
Theory still has a very important part in each, though, as slightly different timings, or harmonies, will need to obey certain rules, usually.Of course, sometimes tunes are pushed and pulled so much that they become unrecognisable, but that's another story...
For some reason, comments can't be left, but for my first para., I was trying to point out that for classical composers who wanted their works to sound the same as they wrote, they needed to put in as much information for the player as possible, and if all that was adhered to, each performance should be pretty similar.