I'm planning to buy one of the above, and, leaving aside image quality, lenses, and other issues, how do the 2 bodies compare in terms of handling to each other as well as to the 450D (Rebel XSi) I'm currently using?
This is highly subjective so the best answer is for you to try them out in a store. If local stores do not have those, you can at least get a feel using the another close model for each brand like the Canon 7D and Nikon D300S.
Obviously both cameras you are looking at are high-end units which dual control-dials, plenty of external controls. In terms of handling, the placement and shape of those controls will make the difference.
For example, one area where I prefer the Nikon is the control dials, having them both high up on the camera lets me control it without shifting my grip. The Canon has its rear control dial lower on the body which for my hands feels awkward. If you use a hand-strap rather than neck-strap, the lower control dial feels even more inconvenient. Also the Canon's joystick is too small and overly sensitive. Then there are subtleties like the Canon's AE-L is sticky and does not let go when pressed again if you change your mind. If you use long exposures often, you may also prefer the Nikon because of the built-in viewfinder shutter.
Now, I am sure someone can say different things that favor the Canon. I manage with my Canon DSLR too ;) but, for the features that I use, the ergonomics of the Nikon suit me better and even more that of my Pentax DSLRs (both cropped-sensors, so a different class).
EDIT: Forgot about the 450D part, which I believe is the Rebel XSi on this side of the Atlantic ;)
The move to a higher-end body is always about the handling and you will notice it. In your case you will also go full-frame which gives quality too but among high-end cameras of the same brand, the handling is extremely similar (Ex: D300S vs D700).
The dual control-dials and extra buttons simply let you work faster. Changing common settings can nearly always be done (IIRC except measuring custom white-balance on the Canon) without going through the menu which gives you more time to take pictures and reduces the chances that you will miss the shot.
The bigger bodies are less cramped too, giving a more comfortable grip and you won't have to smash your nose against the rear LCD and wipe-off sunscreen later because you can't see anything anymore.