You already know half of it. The rails on the 550 are wider, but they are also flat. Your 350 has the narrower rails, and they are NOT flat. So to mount the flat bed on your truck, you will need to offset the mounting surfaces to match. The biggest issue will be dealing with the rails being different widths. Depending on how the bed was made, will make it easy, or hard.
Mounting yoru bed, on his truck will be harder. The flat rails on his truck means the varying heights of mounting rails will have to be changed. Again, the mounting bolt locations will also have to be redrilled.
The biggest difference, is going to be in the length of the framerails. My bet is that the rails on his 550 are longer, which means they are going to stick out past the end of your bed. Easy for you to work around, a lot harder for him. There are a few people on here that have done it. Hopefully they will chime in.
I'm curious, do the support rails on the flat bed run front to back or side to side?
'02 F-350 "Babe The Big Blue Truck" 4x4, Ext. Cab, 8' bed, SRW
DP-Tuner 80T, 160cc single shots, GTP38R turbo, Ceramic coated IH up-pipes, coolant filter, 6637, DIY exhaust brake, openair CCV, propane, SBC DD all ceramic clutch, ISSPRO EV boost/pyro/HPOP pres./rear end temp, 4" lift, traction bars, 255/70R22.5 tires, York air compressor, 16 flashy lights, 12 strobes, 12 switches on console, 9 switches on the dash, 7 two-way radios, 5 air horns, laptop, siren, 105 gal. in-bed fuel tank, Kennesaw Mountain front bumper
'86 H-D Sportster - 16" apes, custom paint, custom pipes to fishtails, single fire ignition, leather saddle bags on custom brackets, bagger fender, solo seat, home made luggage rack, all LED rear lights
'66 Bronco "Big Little Monster"