Rounding errors occur when a numerical calculation requires that a number be rounded or trimmed to a certain number of digits.
Truncationerrors, on the other hand, arise when an infinite process is replaced by a finite one. For example, if you use a Reimann sum of two segments on the left with the same width of segments, the truncation error is the error caused by the approximation of a mathematical process. This is because choosing a finite number of rectangles instead of an infinite number of them results in a truncation error in the mathematical process of integration.
It is important to note that, while it is sometimes mistakenly referred to as such, the rounding error (the consequence of using finite-precision floating-point numbers in computers) is not actually a truncation error. This is because rounding errors are caused by rounding numbers, rather than cutting them.