I bought an old Hercules 3-speed that's in process of being assigned to a family member, and I routinely ride a Schwinn 3-speed. A little experience but not alot, so weight my judgement accordingly.

Raleighs are lighter than Schwinns. Raleighs ride a little softer, maybe due to the Raleigh tubular fork vs. the Ashtabula forged bladed fork. Have heard & my experience bears it out - Raleigh has a shorter top tube, shorter cockpit. Raleigh 26x1-3/8 (ISO 590) tires are available in more variety than the Schwinn 26x1-3/8 tire (ISO 597), of which only Kenda still makes one model in two colors (black & gum). I prefer the Raleigh ride but they're rare or nonexistent in my size (24") and the cockpits of the (smaller) ones I've ridden are a little short. Maybe they're your size. Whichever; if you get a 3-speed, I recommend also getting a 21 or 22 tooth cog. (Standard cog is 18). My Hercules has a defect (binding headset) that must be due to a crappy AMF assembly job. Still have to sort that out.

-Rich