The terminology “semisimple” algebra suggests a generalization of simple algebras, but in fact not all simple algebras are semisimple! (Exercises 1 & 5 in Richard Pierce’s book contain examples)
A simple module is a semisimple module is true though.
Proposition: For a simple algebra A, the following conditions are equivalent:
(i) A is semisimple;
(ii) A is right Artinian;
(iii) A has a minimal right ideal.
Thus to find a algebra that is simple but not semisimple, one can look for an example that is not right Artinian.