Yeah, there are a lot of differences here that could increase conversion. The second form actually includes some information about what this message is about, while the first form is completely bare and doesn't clue users in to what they are inquiring about. The first form has a wall of 9 input fields (with no indication of what's required or optional), while the second has 6 with a relatively discreet link for adding an optional seventh. The second has a lot of friendlier language overall, and has the less relevant links at the bottom separated out from the form better, reducing the amount of information that you need to pay attention to when submitting the form.
I think the mad-lib form technique is interesting, and I'd love to see some good A/B testing of just that one change, without all the rest. You could also test reducing the number of input fields, which is also likely important, clearly marking what's optional and what's not, and hiding optional fields behind a link instead of presenting them all at once.