Both buttons on many calculators are a way to clear or cancel an entry. The C button will clear all input to the calculator. The CE button clears the most recent entry, so if you make a mistake in a long computation, you don't need to start all over again.
This feature has been incorporated in calculators from early models. The first electronic calculator released by Bell Punch Co., Uxbridge, England in 1961 were the Anita Mk VII and the Anita Mk 8.
Anita Mk8 had a Clear Register (CR) and Clear Keyboard (CK) button which continued in later models such as Sanyo ICC-0081 which had CK (Clear Keyboard) and CA (Clear All) button. There was also version of calculators with C (Clear) and CI (Cancel Input) button.
We can see a C (Clear) and CI (Cancel Input) button in the following image: