I've had one of these. I'm still receiving it, really, but it's been going on for about 14 years so far. There are advantages, and disadvantages. They are as follows.
The most important thing to learn in your training is to listen. The development of your 'ear' is crucial. Every human, apparently, is born with perfect pitch. This is the ability to sing any given note on request, within reason. It is also the ability to know what pitch any note you hear is at. It is as though you have the instant knowledge of the hertz value of any note. This is a useful skill, and one that some musicians spend their time trying to emulate. Most people, however, lose their perfect pitch. This is why you have to develop your ear. This is about establishing as strongly as possible in your mind the concept of relative pitch. Interestingly, this is something that people with perfect pitch sometimes find difficult. This is useful for, for example, knowing how to find your starting note amidst a morass of others, or for making sure your intonation is spot-on, or for knowing exactly how to sing or play any given interval. These are core skills to musicianship. A classical musical education will instil this in you. This is why most skilled musicians can sing relatively well- they have trained ears.
Another thing that a classical musical education will give you is if not a love, at least an extensive knowledge of classical music. This is a very good thing. Until you realise how interesting, how varied, and how stimulating classical music is, you can dismiss it with ease, perhaps, as dull. This is because it requires effort to listen to. To an extent, you have to learn how to listen to it. It requires effort to see the recurring motifs and the themes coming around again different ways and places. Go and read this, and this, too. What is best about classical music is that it can be so intelligent. Emotion is far easier to achieve. Find some Bach on the net and you'll see what I mean.
Obviously, a classical musical education brings with it and a little application, technical proficiency in the fields of your choice. This is extremely satisfying. There is a critical point where you stop learning because you're told to, and start learning because the music you're playing is wonderful. On the piano this came at around an upper-intermediate stage for me, but the point may vary according to instrument and musician. As a discipline, singing is very high-yield in that respect.
You get a trained ear- you can listen to things acutely
You get to learn about and hear classical music
The satisfaction of playing an instrument for yourself
There are, perhaps, disadvantages. Received wisdom holds that to have a chance of being truly accomplished, you have to start playing your instrument young. By which I mean between three and seven years of age. This is not always a good thing: I was a very lazy five year-old, and the violin became something of a battleground over which my parents and I tussled. My father wanted me to play because an instrument like the violin that does not have fixed intonation (ie, you have to be in tune- the violin does nothing for you, unlike, say, the piano, where you can't be out of tune)(yes, yes, pedants, I know you can, but if you are it's not your own fault, is my point, now shut up) develops a much better sense of intonation. I didn't want to play because I didn't ever enjoy it. The end result of this is that I've not touched my violin for months.
There is evidence, however, that there's no need to start that young: a guy at my school started playing the piano at 13 and at 18 played Rachmanimov Piano Concerto 1. It's in part to do with what mentality and diligence you bring to your instrument. So don't worry! Don't be sad because you never had the opportunity. With enough application, anyone can succeed.
One real disadvantage is that when your ear is trained and you go back to your old music, you might not like it. You might find it boring, or simplistic. One of my favourite tracks of all time, Belle and Sebastian's Dream of Horses has been ruined for me because I realised that the recorder solo is consistently flat. Seriously. Also, you end up sounding as though you consider yourself superior when explaining the pros and cons to laymen. That's another bad thing.
Other disadvantages: it costs money (did you expect it to be free?), it costs time (ditto), it requires patience. All these apply when learning a new discipline.
Evidently, I've not presented the most objective of views here. My POV is that of someone who has been lucky enough to take this privilege for granted, and who has responded positively. You might not. Insert all-purpose disclaimer here. You will not die more slowly, or prettier, or with a better physique because you had a Classical Musical Education, but it will make the interim more bearable.