no time to look much but chesk this out

To the best of my knowledge, AASHTO now uses the ASTM A709 Grade HPS 70W designation. ASTM A709 uses the dual designation Grade HPS 70W/HPS 485W. I have explained the modifications to these specifications below. Please be aware that the ASTM A709 Grade HPS 70W/485W specification applies only to the Q&T product at this time . The TMCP product is currently being balloted by ASTM, but is currently not included. When an owner desires to allow use of the TMCP product, special provisions should be included in the contract documents.

http://eldau.ru/pdf/PART_3.PDF

Officially, those letters are just a piece of the grade designation and don't actually mean anything, which is why you won't see them explained in the standard.  Practically, they are a handy mnemonic for the type of steel.

"W", as you said, stands for "weathering".  A 709 Gr. 50W is equivalent to A 588 (except the A 709 carries some additional Charpy testing requirements).

"S" stands for "structural".  (I thought it oughta stand for "seismic", but no one asked me.)  50S is the equivalent (plus some Charpy testing requirements) of A 992, structural shapes.  It has a cap on ultimate strength and on yield-to-tensile ratio.  My understanding is that this is to allow reliable design for "strong column & weak beam".

"HPS" stands for "high-performance steel".  The HPS steels are characterized by higher toughness than other steels of similar strength, and the 70 & 100 ksi grades are more weldable than the non-HPS equivalents.

The HPS grades come in both Q&T & TMCP, per ASTM A 709.  Availability is a different story.  Last I knew, TMCP had a thickness limitation of 2 inches.

Hg