I've talked before at length about both Raleigh and Schwinn three speed utility bikes. Let's consider the common, cable brake model Raleigh Sports style bike, and its Schwinn counterparts. We're talking bikes with diamond or ladies frames, 26 x 1 3/8 or similar wheels, Sturmey Archer transmissions, and accessories befitting a basic utility bike. Let's exclude the higher-end bikes with Reynolds 531 tubing or Schwinn Cro-Mo tubing.
So this leaves us with the basic 3-speed bikes: the Raleigh Sports, and the Schwinn New World, Schwinn World Varsity/Traveler, and similar baseline 3-speed bikes. Here are a few key differences.
Raleigh Sports bikes are brazed and lugged, made from mild steel tubing. They have around a 72 degree frame angle and place the rider in a sitting-up type position. The Schwinn three speeds have somewhat more relaxed frame angles. Early Schwinn 3-speeds from before 1945 are fillet brazed of seamless, mild steel. Later bikes are flash electroforged welded of mild steel. The fillet brazed frames are a little lighter.
If you prefer a larger cockpit and more relaxed frame, look for the Schwinn. If you prefer tighter frame angles and lugs, go for the Raleigh.
The Raleigh frame is lighter than the electroforged frame of the same size, but heavier than the fillet brazed seamless frame of the same size. So an early Schwinn is lightest and, in my experience, most responsive. The Raleigh frame is a good balance of weight, responsiveness, and durability. The Schwinn electroforged frame is really bomb-proof, but also heavier than its needs to be. The later electroforged frames from the 1960s-70s have a very heavy, "dead" feel compared to earlier Schwinns and the Raleighs. Electroforged Schwinn frames from the 1960s-70s do make great commuters - cheap, durable, and not particularly theft-prone.
I consider the frame, the most important factor on this list, because of the angle differences, weight differences, and differing feel.
Raleigh bikes generally have cottered, three-piece cranks. 48-tooth chainrings are the norm, but some bikes have 46-tooth chainrings. Early Schwinn bikes from before the 1950s can have one-piece cranks or cottered three-piece cranks. Later Schwinn 3-speeds from the 1950s-60s have one-piece cranks. The Schwinn bikes usually have 46-tooth chainrings.
If you prefer very simple bottom brackets at the expense of a little weight, the Schwinn one-piece is fine. If you prefer cottered, look for a Raleigh or an early Schwinn.
If you want a very wide variety of pedal availability, go for the Raleigh. If you prefer American, cruiser-style big pedals, look for the Schwinn.
I've fonder Raleigh frames, especially the bonderized ones to resist rust better than Schwinn frames. If you live in an area where frames like to rust, the Raleigh has a slight advantage here.
As to the fenders- Schwinn stainless fenders are most rust-resistant. Raleigh fenders are usually pretty good, but do rust without proper care. Some early Schwinn painted fenders rust fairly quickly. I have to spot-clean my 1947 New World fenders of pop-up surface rust every spring.
Tires and Rims
It's easier to get a variety of tires for the Raleigh rim (ISO 590mm). The Schwinn size leaves you with basically just Kenda tires as an option for good tires, but the Kendas are pretty good still (ISO 597mm). I think the rims on the Raleighs are a bit lighter than their Schwinn cousins, at least comparing the Westrick to the S5 and the Endrick to the S6. The Schwinn rims are well-made usually, but a bit heavier. The Schwinn S5 rim, I think, is a real dud. The S6 rides nicely. The Raleigh Westricks are a bit heavy, but very durable. The Raleigh/Dunlop Endrick can be a very nice balance of durability and reasonable weight.
If you want to replace rims with aluminum, the Sun CR18 is a direct swap for the Raleigh size, but the Schwinn size has no direct replacement that I know of.
Brake Calipers and Cables
Earlier Raleigh use proprietary, double-ended cables. They work very, very nicely when the cables are good, but finding good cables today is a pain and expensive. Later Raleighs use a standard cable, but the calipers are not as good. Schwinn brakes are better usually - earlier Schwinns use US-made steel calipers taking standard cables.
Later Schwinns use Weinmann alloy brakes that also take standard cables. The Weinmann Schwinn brakes are especially light and function fairly well. The Schwinn has the advantage here. Some very late Raleigh Sports also use the Weinmann alloy calipers.
Saddles vary all over. The Raleigh stock Brooks B72 is a good saddle. The Superbe stock B66 is the best saddle model ever produced. The 1930s-50s Schwinn horsehair tourist saddle is very nice, but usually they're beat when you find them today. The rubber mattress saddles from the 1960s-70s are OK but not terribly comfortable. Raleigh eventually swapped to the rubber mattress as well. Do yourself a favor and buy a new Brooks B66 or B72 if you're serious about riding.