How does the Boeing 737's implementation of RNAV and VOR approaches compare to Airbus?
An RNAV approach in the Boeing 737 is usually flown in LNAV/VNAV mode, meaning the Flight Management Computer (FMC) provides both lateral and vertical guidance. A VOR approach can be flown by manually dialing in the VOR frequency into the NAV radio and dialing in a course, which can then be tracked using VOR/LOC mode. Vertical guidance may be provided by VNAV, but the pilots can also manually dial in a vertical speed (V/S). However, the 737 can also fly a VOR approach in LNAV/VNAV mode, meaning the same way an RNAV approach is flown.
On the Airbus A300 and A310, the VOR approach works quite similar to the 737: you can manually dial in the VOR frequency and a course and follow that using NAV and V/S mode. Alternatively, the VOR approach can also be flown in profile mode, where the FMC provides lateral and vertical guidance (no need to dial in the VOR frequency any more). The same profile mode is also used for RNAV approaches, which allows flying RNAV and VOR approaches the same way.
Starting with the A320, Airbus removed the option to manually track a VOR. The only possibility to fly VOR approaches is to use managed mode (the new term Airbus uses for guidance provided by the Flight Management and Guidance Computer, FMGC). The same mode is used for RNAV approaches (if flown using LNAV/VNAV mode, not true when using LPV using an SBAS, but I don't think that's what you are asking about). All non-precision approaches (NPAs) are flown this way, so there is no difference between VOR or RNAV approaches, however the FMGC may (depending on equipment) rely on the VOR data (see related incident).
Both Boeing and Airbus offer another way to fly NPAs though (regardless of VOR or RNAV or whatever NPA approach), which is more similar to a precision approach. Boeing calls this Integrated Approach Navigation (IAN) and Airbus calls this FMS Landing System (FLS). The indications on the PFD and the procedures to fly this approach are then very similar to an ILS approach: