OK, on the basis of somewhat limited information, I'd suggest that there is a problem with your cassette player(s) and with the Sony cassette.
The player may have a slight problem with the motor, or the rubber band connecting the motor to the drive mechanism, and it's losing a proportion of its power. The SONY tapes may have lost a bit of their lubrication, and they need a little more effort now to play. When the machine is in FF or RW mode, ALL the power goes to those functions, and there is more power, so this works fine. The TDK tapes do not have the lubrication problem, so they work either way. When you try to play the SONY tape, some of the power goes to drive the capstan and some goes to pull the tape onto the take-up spool, and the machine struggles, it senses End-of-Tape and stops.
If this is the case, I'd expect that hand winding the SONY would feel slightly stiffer than doing the same with a TDK, but you didn't answer that question. If you can use a more robust cassette player, then both tapes may work OK.
I'm just trying to fit the facts (?) to a hypothesis. The above seems to fit.
I'd suspect that the SONY tapes are NOT 'new' - do they show a date of mfg. They may also NOT be mfg by SONY, but maybe in China (or somewhere else in SE Asia) under licence. Either way, the tapes may be OK, with a machine with more power.
Geoff