In the UK any VAT-registered business is required to show prices that are inclusive of VAT (unless they're advertising primarily on a business-to-business basis, where the purchasers of their goods and services are also likely to be VAT-registered). So any hotel price quoted to you by a UK business must include the VAT that's charged on the booking.
However things are different in the USA, where it's normal for 'local taxes', etc to be added onto the advertised price when you make a hotel booking. As Reservations.com is based in Florida, it might be that they're quoting VAT-exclusive prices on their website and then adding on the VAT afterwards. Further, I note from their website that they charge your card in US dollars, rather than in sterling, which could further add to the costs you have to pay when you book through them.
Reservations.com has an appalling reputation in the UK, with 61% of TrustPilot reviews awarding it just 1 star:
https://uk.trustpilot.com/review/reservations.com
Things are even worse on their home country, with 77% of reviews on Sitejabber rating their company as only worth a single star:
https://www.sitejabber.com/reviews/reservations.com
Your chances of getting a refund from a US company which clearly doesn't care about its reputation are probably nil. Since your contract is with Reservations.com, Days Inn aren't obliged to offer you any refund. However it might be worth complaining to the company about their general policy of allowing their rooms to appear on the Reservations.com website (thus resulting in potential customers paying more than they expected), using your own booking as an example. That might well result in an
ex gratia payment from them or, possibly, the offer of free accommodation at another time.
'Days Inn' is actually a trading name of Roadchef. I suggest using good old-fashioned snail mail to write a letter of complaint to . . .
Mark Fox,
Chief Executive,
Roadchef Motorways Ltd,
Roadchef House,
Norton Canes Motorway Service Area,
Betty's Lane,
Norton Canes,
Cannock,
WS11 9UX.