There are a couple things you'll notice right away:
1) Start-up time will be quicker
2) shutter-lag: the time from when you press the shutter button until the time the photo is taken will be shorter in-general
3) pictures at higher ISO will appear less grainy
On the negative side you'll notice that the amount of subjects in a given photo that are in-focus will be less. If you take the same photo, framed the same way with your old camera and the 650d you'll see that more of the 650d photo is out-of-focus. what should be in-focus is what the focus point(s) were on but things in the background will be less in-focus. This is desirable to many people but not all.
4) if you use the "sports" mode on the new camera it will do a dramatically better job than your current camera - at least within the limits of your lens and distance to subject. By that I mean you can't expect to use a 55mm lens and track a kid playing soccer 40 yards away and expect great shots.
After that, the differences are more of potential improvement. You have the ability to attach an external ETTL flash - which provides a quantum improvement in flash shots vs the built-in flash of ANY camera.
You have the ability to use different lenses which are specialized and thus produce better results than a built-in lens or kit lens. For example if you like taking photos of birds you could use a 400mm f5.6 lens.
The challenge becomes: do you want to buy those other lenses as you determine what you need? And, very significantly - are you OK with the added bulk a DSLR requires? I've shot with DSLRs for years. But several friends and family members have tried and gone back to using digicams and/or cell phones because those are with them and they never wanted to carry around the bulk of a DSLR.