I always called them Sheffield Wednesday and Sheffield United in person, Sheff Wed and Sheff Utd if typing it out.
Sheffield Wednesday were the more dominant of the two in my youth but they haven't been in the top flight this century. Jesus, I'm old now! Sheffield United haven't exactly ever been a mainstay, only sporadically being up there in my lifetime.
Sheffield Wednesday have actually been decent in my memory too, Sheffield United never have. Was at Wembley when Sheffield Wednesday beat us in the League Cup final in 90/91, they made both cup finals in 92/93 and they've had more notable players like Des Walker, Chris Waddle, Trevor Francis (player/manager), and David Hirst who Fergie has gone on record as wanting to bring to Old Trafford before Cantona became available by a stroke of luck.
They also finished 3rd in 91/92, Sheff Utd haven't got shit on any of that! Their only claim to fame was Brian Deane scoring the first ever goal in the newly-formed Premier League against us.
Can't go as far to call Sheffield Wednesday "Wednesday", that's just too familiar and can be reserved for the people of Sheffield.
Edit: Random thought, do Sheffield United fans in Sheffield call them "United"? I think they should. Same for West Ham fans in London and Newcastle fans in Geordieland.
Another thought: Sheffield Wednesday I see as a bigger club not just for all the above, but also because FA Cup semi finals were often held at their place due to them having a large stadium in those days. It's infamous now because of the tragedy that took place there. Still, Sheff Utd can never be called "Sheffield", it's just wrong in my eyes!
I also have a bit of a soft spot for Sheff Wed on the grounds that we were poor growing up. I desperately wanted a United kit in the when I was a kid and told my mum I wanted a football shirt, thinking she'd know what I meant. She managed to find me a 78/79 Sheff Wed home shirt and a random Sunderland 3rd shirt from about '83 in the Barnardo's second hand shop that I'd wear to primary school football practice around 1989/1990. Good times!
In conclusion, bog off calling Sheff Utd "Sheffield". They're the smaller Sheffield team even if I do like their "chip butty" song. Always was, always will be!
Old man yelling at clouds gif.