The phrases are practically identical in meaning. "Over the weekend" is popular in American English. The only slight difference is that "over the weekend" refers to the entire period, whereas one might use "during the weekend" to refer to one particular instance at a particular moment. "I am going to Paris over the weekend". "During the weekend, he lost his wallet".

during the weekend, has a connotation for a beginning point from where you continue, or as the weekend was happening. over the weekend, has a more non-discript feeling to it, "do it as some time, but within the weekend". As Randy said very nicely. Give him the best answer. :)