The similarities and differences between the two depend on the model years and manufacturers.
For my 10yr Dodge, the difference between the 3/4 and 1T are the rear spring packs. The wheels and tires, axles and brakes are identical. My 9k GVWR truck has a ~1800 payload rating, and i've had 2500 in the bed and totaled out at 9980lbs; was a little squishy on the rear end but otherwise fine.
With the Fords, back in the 90's the 4wd 250 got the Twin Traction Beam with leafs front suspension while the 1T got a solid axle with leafs. The 2wd units were all TTB with coils. The rears were the same axles, same wheels and tires, and different spring packs and blocks. As the years progressed, the 4wd 250 got a Dana 50 solid axle while the 350 got a Dana 60 solid axle.
Fast forward to today and there are even more differences, like the Dodge 2500 has a 5-link coil while the 3500 is on leafs.
For a few year old truck, the ride is going to be similar between the 3/4 and 1T. The overloads take a couple inches of sag before they are engaged, so the two truck will ride pretty much identical, in most cases.