I've personally never heard labour being used as a verb unless it's used with another word (e.g to perform labour), it's commonly used as a noun. I kind of agree with what you said about getting paid / not getting paid, but not exactly. Work is a more general term, so it can refer to various types of work, including your job, school work, housekeeping etc. it can be light work or happy work or personal work. But labour is more specifically an official / work based term referring to hard work and manual work, it has a somewhat negative nuance. People doing labour are usually overworked.

The differences are in spelling only. The British and other previous colonies spell it Labour while the Americans spell it Labor. There is no difference in the meaning. Another example is the word Colour used in the UK while in the USA it is spelled Color.