The difference is in the sound, the pixels and compatibility. I'm oldskool so I like to play nice old games like Paradroid 90, Datastorm, Gravity Force, Master Blaster, Rick Dangerous etc etc. As a matter of fact I hooked up my trusty old A500 and played a couple of those game this very day. If I want to play something heavy like F1GP I whip out my '040 Amiga.
The advantages with the real thing:
1) The sound. The WinUAE emulation is fine but it just doesn't sound like the real thing. I fiddle around with trackers a bit and I notice the difference. I also had trouble playing .mods that I downloaded, all but Protracker 3.00b+ hung. So...
2) The pixels. There just is no way to emulate the saturation of my 1084. If I emulate with raster lines the colors just die. If I don't, I'm left with much too sharp boxy pixels. Also, 50Hz on my Dell 19" Trinitron flickers a lot more than my 1084 @ 50 Hz, probably due to the 1084's phosphor layer which retains the picture for longer. (high-end) PC monitors just aren't nice to look at when using too low Hz. Also, copper shades that were perfectly smooth on the real Amiga suddenly look bad.
3) Compatibility is MUCH better for these nice old games and utilities from the early 90's. I'm having a very hard time running certain utilities in WinUAE, even though I configure it like the real machine. When I run it on the real thing with the exact same setup as in WinUAE, the same software that hung every time runs perfectly every time. This might of course get better with time.
There is of course a big advantage with emulating too - no disk swapping, no copying files or disks (much easier to drag and drop and organize on the PC), and you can download huge amounts of stuff to try out. It's also a lot nicer to code assembler in the emulator, you can start two emulation windows and code in one, start the program in the other, if one hangs, you still have your source. You can size the window as you like and enjoy a 100Hz+ picture, etc etc.
Why not do like me, enjoy the best of both worlds?