They are both past tense and have similar meanings. The expression "used to work at the company" emphasizes the fact that something changed and I do NOT work there now. The form "used to do" (expression) generally gives us more information than "I did" (past simple).
For example:
I smoked. (Yesterday? For 20 years? Once? In the past? It could mean any of those!)
I used to smoke. (means I smoked regularly, but I quit smoking and now I do not smoke anymore.)