Depends on what you call a difference. The nice lady at the DMV, or the IRS or the county assessor's office will likely know far more about you than you ever will about her because of access to data. A lot of that data is not publicly available.
What I mean is the people at the DMV, IRS, and county assessors are total strangers and have no connection to me. Giving them access to my information, publicly available or not, is no different than handing it out to any arbitrary members of the public.
I have no reason to trust them any more than I trust anybody else. Saying, "They won't misuse the data because it's their job not to," is as comforting as "Nobody will misuse the data because it would be illegal."
I think the difference is more that with FOIA-style public lists, that people who will misuse your data can look you up directly, whereas the clerk at the DMV actually is a total stranger who if anything cares about you personally even less than a total stranger might.
It's a safe assumption that if the government is collecting personal information then random strangers will have access to it.