The biggest difference from a riding point of view would probably be the geometry. If you are worried about having a proper track bike then a conversion - no matter how nice - may not be the ticket. If you just want a new bike then nothing will do other than a new bike.

I reckon that fixing up a nice old frame like that would be cool - you get a unique bike that you know you like and take a small stand against buying pointless new things that you don't need (of which I am always guilty)..

You could upgrade parts on the current frame and then, if you decide you want a new frame, transfer the parts across to that.