If you get a rise, you have an increase in the amount of money you earn. If you get promoted, you have a higher rank in your job.

An alternate meaning for getting a rise is 'to make someone upset'. We would usually say 'getting a raise'.

In the USA, I have never heard "getting a rise" in the context of a pay increase. The two phrases have different, but related, meanings: "The company is very profitable, so, that means that I'm getting a raise!" A raise is an increase in salary or wages. It could be merit-based, which would often result from a "performance review", or it could be a cost of living raise, which has nothing to do with individual performance. So, a raise does not necessarily mean a promotion. "The boss really likes me, so, I'm getting a promotion to Vice President!" A promotion is a change in responsibility and/or title. Ideally, a promotion should also include a merit-based raise, but some companies cannot afford raises or give title promotions without raises as a way to inexpensively motivate employees.

Thanks everyone for the comments! I saw the phrase "getting a rise" in a British English textbook, so probably the difference in the spelling depends on that, as Susan and Carlos suggested.