 
Eric K. answered  08/30/19
PhD in Organic Chemistry with 7+ Years of Teaching Experience
Great question! This was actually a hotly-contested area of research for many years, but the succinct answer is that they differ in how electron delocalization occurs to stabilize intermediates.
In greater detail, non-classical carbocations are stabilized via bridged transition states, often via 3-center-2-electron bonding. This is in stark contrast to a classical carbocation, which might be stabilized by simple resonance. A quintessential example of a non-classical carbocation is the 2-norbornyl carbocation, and a benzyl cation is a great example of a classical carbocation.
Why do we care? The existence of non-classical carbocations has profound implications for chemical reactivity, and reaction kinetics can be drastically altered.
 
     
             
                     
                    