Are idempotent functions the same as pure functions?

An idempotent function can cause idempotent side-effects.

A pure function cannot.

For example, a function which sets the text of a textbox is idempotent (because multiple calls will display the same text), but not pure.
Similarly, deleting a record by GUID (not by count) is idempotent, because the row stays deleted after subsequent calls. (additional calls do nothing)

A pure function is a function without side-effects where the output is solely determined by the input - that is, calling f(x) will give the same result no matter how many times you call it.

An idempotent function is one that can be applied multiple times without changing the result - that is, f(f(x)) is the same as f(x).

A function can be pure, idempotent, both, or neither.

No, an idempotent function will change program/object/machine state - and will make that change only once (despite repeated calls). A pure function changes nothing, and continues to provide a (return) result each time it is called.