I should point out I'm not much of a shopper - I'll tend to go to shop to buy something specific, rather than "just pop by to browse". That said I'd prefer to go shopping at Fortnum and Mason (Piccadilly) rather than Harrods (Knightsbridge). Both are high-brow places, both sell stuff you don't need, and both have ranges of their own unique products (both also have an ice cream parlour nowadays too).
Visiting F&M is an experience - the hat and tails wearing doorman will open the door with a friendly welcome for you as you approach. On entering, the shop is grand - it has high ceilings, is decadently appointed and has sumptuous products everywhere. The whole thing is a spectacle, a celebration of shopping and choice with all of the wares cherished and special (even if they're actually quite ordinary). If like me you have little shopping stamina, they have a cute bar down the beautiful spiral staircase where you can get a glass of wine and a nibble or two. If that's not what you want, then there's the ice cream parlour or various restaurants. The shop isn't usually over-crowded, and is generally frequented by considerate shoppers who conduct themselves with a little decorum.
Contrast this with Harrods. The doormen are so over-worked they'll probably just be lurking about the entrances whilst an inconsiderate, tired, irate and rude fellow shopper lets go of the swing door in your face. You'll enter the fairly run-of-the-mill shop and observe the over-priced ordinary products, with the odd interspersed Harrods own-brand facsimile of the products you know and love (although to be fair, Harrods do have some lovely products too - you just have to search for them). The ceilings are low, the spaces cramped and busy. There's almost no spectacle, except maybe the Egyptian escalator or a few other choice areas. There are also some places to sit down and take a moment to regroup. These are fancy versions of canteens, with bustling service a constant rotation of customers. All in all, it's basically an expensive supermarket. If it wasn't called Harrods, you probably wouldn't go near it.
So all in all, I have to say, F&M is by far the better option for me. It's on my list of "tourist stuff for visitors", and since it's a place Mrs Cat shops from time to time, it's had a bit of my custom too. In the Internet age, one wonders what the future holds for physical stores, but I'd guess the likes of F&M will do just fine. They offer something you can't get anywhere else, and you certainly can't get online. I'd imagine we'd all like the idea of having an experience you can't get anywhere else, and buy unique products you can't (easily) get online - even if they're more expensive than the equivalent chain-store products we're more used to. Time will tell...