I don't know about quality and construction, but I would pick the weymouth. It has a more elongated proportion which I like. Also, the toe on the Westminster is a little boring. The finish does look first-rate, on the other hand.
Dr Bresch has a pair of Alfred Sargent Shaftesburys, which is just the Westminster model in a different colour. Â His posts and pics on Ask Andy's: herehereherehereand here. Also, I'm pretty sure Shipton & Heneage's wholecuts in their Premium range are exactly the same shoes (even down to the names for most of the models.) so it might be easier and/or cheaper to get the shoes from them instead.
I can't really comment on the AS's as I haven't seen them in person, but my C&J Weymouths in dark brown antique arrived an hour ago and let me say... Wow. I am impressed. This is my second pair of handgrades (first are Aintrees in beige) and I recommend them wholeheartedly. Also, P Lal is highly recommended. Less than $400 usd delivered. The prices cannot be beat and the service is exceptional.
I have the Salisbury and have never seen the Weymouth in person. But with that caveat, if you have the money, buy the Weymouth. Its last and finish look more elegant to me than the AS wholecuts. My main complaint about the Salisbury is that it is a fine shoe, especially for the money and after its first polish, but it has a clumsy last that is just not to my taste, as do all AS's premier models (so far as I can tell).
Is the weymouth design considered more or less formal than the typical captoe? I know the medallion on the toe would be considered less, but what about the otherwise plain nature of the shoe (not exactly sure what the lack of closed/open lacing is referred to as).