How much fuel do the new A320-Winglets burn vs. the B737-800?
Airbus says this about their new blended winglets (aka "Sharklets"), being fitted to new A320s, and retrofitted:


  Airbus’ next step is the large Sharklets™ wingtip devices, representing another element in the ongoing continuous improvement programme for its best-selling A320 Family. These devices improve aerodynamics, reducing fuel burn by up to 4 per cent – which amounts to annual savings of more than 900 tonnes of CO2 per aircraft.


Given that an A320's fuel burn rate is approximately 5.13 gallons per seat per hour, reducing this by 4% gives you 4.92 gallons/seat/hr.

Comparatively, a 737-800 burns about 4.88 gallons/seat/hr.

Source:


  A 737-800 burns 4.88 gallons of fuel per seat per hour, compared with the comparable A320's burn of 5.13 gallons per seat per hour, according to The Airline Monitor, an industry publication.


So they're near enough the same.