I've been shaving with a traditional double-edge safety razor for about 20 months now. It's good. There are a couple of reasons I switched.
The main one is the insultingly exploitative business model of the shaving gear manufacturers. Since the late 1960s they have been ratcheting up the complexity and cost of razors in order to extract more profit from their customers. It's a classic example of patent-driven innovation: they have to keep coming up with new gimmicks that they can monopolize, and use bulshytt to convince people to buy these "better" razors instead of choosing on the basis of objective value or quality. This process has been obviously ridiculous practically since the introduction of cartridge razors, and has long since passed the stage of blatant self-parody.
When Gillette introduced the Mach 3, I stayed with the Sensor; a few years later I decided to see if a Boots own-brand Contour clone was any good despite being a lot cheaper. It turned out than any difference in the razors was dwarfed by variance in my shaving technique, so I switched to the cheaper one. There are now other options in the "less eye-watering than Gillette" segment of the market, such as the King of Shaves Azor or the Dollar Shave Club, but they still buy into the Trac LXXVI bulshytt.
The secondary reason was that although Boots own-brand razors do the job, they are a bit crappy and ugly. I had a vague desire for something more elegant which involved chucking less plastic in the bin. Partly based on satisfied reports from Tom, I invested £40 in a new old-fashioned razor and some consumables.
The Edwin Jagger DE89L is a lovely object. It is made in Sheffield from chrome-plated brass, and has a nice heft: it weighs 76g which is more than four times as much as my old plastic razor. It has a wonderful economy of design (Occam would approve) with only three parts each of which serves multiple functions. The handle is threaded to screw the blade clamp together; the bottom of the clamp includes a safety guard; and the top of the clamp acts as a guide to the angle of the blade against the skin. It's just the right shape for safely shaving under my nose.
ETA: For blades I'm currently using Feather Japanese blades, mainly on the basis of hearsay and prejudice, er, I mean their reputation for quality and sharpness. I don't think it's possible to make a meaningful comparison without fitting different blades to identical razors and using them both during the same shave, repeatedly. (See above about variability of technique.) And I only have one DE razor at the moment.
I've also switched from using a shaving oil to a shaving cream and badger brush. Taylor of Old Bond St, Court Hairdressers are awfully posh but a 150ml bowl of their shaving cream costs less than £7 and you only need one or two ml for a shave. More fun to use and easier than oil to clean up afterwards.
So now my morning shaves are still inexpensive but much more luxurious. Very satisfying :-)