The use of "the restaurant" in the first sentence signifies thy the restaurant is known to the speaker whereas in the second sentence the use of "a restaurant"' tells us that the restaurant is not known to the speaker.

Hello Agata, In the first sentence He was working at the restaurant for a month last year We know which restaurant he was working at 'the' and the connection is to emphasis that he had a job at a particular restaurant for a particular time. The second sentence He worked at a restaurant for a month last year. This is completed action in finished time and we don't know which restaurant. (a restaurant). We are more interested in the completed action rather than detail.