The difference is perhaps best illustrated by the fact that you can elicit the truth from the evidence, but not solicit (soliciting is asking, implying a "conscious" entity you're seeking something from). Relevant definitions from OED...
elicit - to bring out, educe (principles, truths, etc.) from the data in which they are implied.
Also, to extract, draw out (information) from a person by interrogation; sometimes with object clause introduced by that.
solicit - to seek after; to try to find, obtain, or acquire.
As implied by the second part of the first definition, in OP's exact context either word could be used. And I would say the intended meaning is so clear it's pointless to suggest the actual choice of verb could have any semantic significance there (most people would just use get or ask for anyway).
That OED definition implies that when you elicit information from a person, you're treating him insensitively, inhumanely. The other two current answers say it's when you get information out of someone surreptitiously or indirectly.
I think all these implications arise naturally from the fact that you normally elicit information from data/things. If it's from a person, you've effectively "objectified" him, which can lead to a range of negative connotations (but none universally observed).
Note that there's a much clearer and more consistent distinction between the two words in, say,...
1: We will solicit feedback from the users
2: We will elicit feedback from the users
In #1 we're going to ask, but we might not get the feedback. In #2 we're saying we will get it (by force if necessary! :).