Here's my take on the story.....
The T.1 already had provision and clearance for a range of Practise bombs, 1000lb bomb and other weapons.
The "War Role" T.1A conversion added 3 servos for additional weapons: BL755, Sidewinder, plus 1 (Matra155 rocket pods?) and these were allocated to the TWUs.
After a few years of operations the RAF realised the aeroplanes at the TWUs suffered high FI usage, compared to the high flying hours but benign flying at the FTS (at Valley), hence had to deal with airframe management of 2 separate fleets.
The T.1W was a Service Engineering Mod (Not a DA approved Mod) to add 2 of these weapons to the T.1 (one required additional server with wiring down the wing leading edge to the pylon, so could not be installed without a lot of work) to give the fleet a wider weapons capability, especially as they then became centred at Valley.
I don't think there are visible differences in the aeroplane and as far as operation is concerned, there is only a minor weight & cg difference (masked by all the other mods & repairs over 40 years!)