No news yet on when the Record Office will be re-opening and nothing yet to even say if the staff are still working from home or have returned to do their usual duties but not yet open to the public.
I looked on Durham Records online, again just transcriptions, but both records are there and both say 15th May 1737, and the one in Tanfield says 'Katherine' was of Medomsley.
It 'may' be that the transcription you have has the wrong year for one of them and one is indeed a banns record (or an entry saying a parishioner married in a different parish)
Durham Record Office catalogue shows they only hold actual banns registers for both of these parishes from 1754 onwards :-(
When the archives do open, if no-one is able to look it up for you, then an option is to use their 'quick search' service. Order online (£5 per record) and get the image delivered by email.
I think you may well be correct in thinking both records are for the same people, but I can't offer to look it up at Durham as I am nowhere near the area.
If you get to the bottom of it do post back. I like to know how things turn out (I'm nosey!)
Boo