Some of this will be just rehashing the rest, but I'll toss in my buck-fifty, since 2 cents don't buy squat...
I have a cheater. I check his stats and everything.
(I reroll his stats on my sheet and adjust everything so I can adjust the AC's and DC and damage appropriately-He doesn't think I've caught on yet)
I have a rather expansive software library, one program in particular lets me keep track of and update the PC's enmasse and adjust it subtly. It's kind of fun to do, and since the program prints me out a nice sheet with the updates and changes in red, it makes it easy for the PC's to adjust things.
We play Ravenloft at times, and me having a copy of the PC sheets make it easy to do subtle checks, and nothing makes a player sweat more than to have me look at the sheet and apparently make a change.
I've had a player accidently forget to add a feat, make a mistake on skill points, miss a stat adjustment, etc.
I write all my adventures in-house, so it's nice to have them for reference.
In our "Hard core-immersion" game, the players have thier equipment list and thier appearance written down. Mechanics are my problem as the person who runs everything. They just play, I make the rolls. It's run, they like it.
Most of us have children, and nothing ruins a gaming night quicker than a toddler with a 15th level wizard in her mouth running around the house. When said toddler is caught, she jams the wizard in a diaper full of poop. What a horrible end for that PC. Luckily, I had a copy of the PC sheet, so the PC was saved from toddler-feces-death.
Ever asked a player for a spot or listen check, and the ambush is basically blown right there? I might as well have my goblins wearing neon pink spandex and lime-green mohawk wigs! So a lot of times, I roll thier Spot/Listen/Hide checks for them.
(Another cute trick I've done since I started DMing Ravenloft back about 1989, was to have the players make a bunch of die rolls, and record them on a sheet of paper (in nice columns for d4, d8, d10, d6, d12, and a whole boat-load of d20) that way, I don't even have to roll dice, I just glace at that player's die roll sheet, cross off the next die, and go with it. Makes it easier. I have had players try to hand me sheets of all high rolls, but then I just take the top numbers listed and enter them as the random seed on a program that pumps me out 250 rolls of each die and prints them out, cheating pretty much stopped there)
I've had more than player noticed, when they died, that my PC sheet was different than theirs. It was simple, they'd had things happen they didn't notice (the blessing of a dying diety that they'd aided, the curse of someone they had robbed, a bonus here, a penalty there, loss of XP from this that or the other thing)
Plus, my players don't mind me having a copy of thier PC sheets. They know they can trust me.