If advertising is an art, direct marketing is a science in which key components are tested, evaluated and run again at higher, more profitable rates of response. I like to compare general advertising design vs. direct marketing design to a Masarati vs. a lawn mower. The lawn mower may not make it into the Museum of Modern Art, but it cuts grass. Direct marketing has to cut the grass. Information flows from the advertiser to the prospect.
Direct response is interactive or as some prefer, proactive.
In direct, we want to get information from the prospect. In lead generation, we point-blank ask for the information we need. There are always exceptions to these parameters, of course, but basically, those are the differences. Also be sure to ask to see the pieces they tested against that mailing. If they can satisfy you on that score, great.