The CMOS (with the C in the middle) and NMOS (without) parts are totally interchangeable as long as software goes. Manufacturer doesn't matter either.
One difference is that the power saving modes of the CMOS controller are not implemented in the NMOS version. This doesn't matter if the original design uses an NMOS part.
The reset pin works differently, but again, it doesn't usually matter when putting a C part in place of an NMOS chip. It could be a problem, since B&O's reset circuits are often more or less silly (or too clever, depending on how you want to look at it) and don't follow the chip manufacturers' application notes.
The on-chip clock oscillator differs as well, but it won't matter in this case.
The internal architecture of I/O pins differs between these two versions, but contrary to regular CMOS logic, the 80Cxx will accept TTL logic levels directly, so that won't prevent direct replacement either.
Finally, a xx32 will work in place of a xx31, but not the other way round (8032 has more on-chip memory and peripherals).
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What you should stay away from are the further developed versions of the basic 31/32/51/52 controller. They are usually only made by one manufacturer, may require chip specific initialization, and the faster versions may have different instruction timings, which will definitely break any code that uses software timing loops.
That's the list of differences, but can I guarantee that you can replace the original 8032 with a 80C32? Unfortunately not, but I strongly suspect it will work without problems 
PS. The "1980" will be simply Intel's copyright date on the original design, not the date of chip manufacture, or you have managed to find a really old batch.
PS/2. For completeness: the xx51/52 contain on-chip ROM with the program code, which has been mask programmed already during the chip manufacturing process. These chips will therefore be application specific, and it's impossible to replace them with anything but an original spare part chip - or a very expensive EPROM version of the controller, but then you'd need to get the program code from somewhere as well.