When you actually go there, you realise that those 2 addresses are in fact the same building

38 Grosvenor Street is the side entrance, this is where people queue up to deliver and pick up their passports, and is usually locked - 1 Grosvenor Square is round the corner overlooking the actual square, and this usually has a couple of security guards outside so guess that entrance isn't locked.
Anyway we sent ours to 38 Grosvenor Street and we got our passport requests fine, so I'd advise going with that one - but I suspect it doesn't matter.
Wayne.