Is there a meaningful difference between the terms "plain bearing" and
"plane bearing"?  If so, what is it?

I would assume that "plane bearing" is a mispelling that has become
commonplace, but would like to find evidence one way or the other. You
get lots of hits on web sites, even from bearing manufacturers, on
"plane bearings", but just getting hits doesn't inspire confidence.

I can't tell you much about plain bearings, but even if "plane
bearing" is a misspelling, there is a reason you would get a lot of
hits on web sites with the misspelling.  The websites want to attract
as much traffic as possible.  They know the most common search words
and phrases people type in when they are looking for a website like
theirs, and they provide links to these terms.  Even if one of the
terms is a misspelling.

I don't know if any producers have two websites using the different spellings.
Plane can hardly be considered just a mispelling:  7+ million hits, 
12+ million for "plain".  I can see a logic for both:  Plane: 
plane/smooth bearing surfaces  - albeit curved, of course, and
sometimes spherical;  Plain, not roller or ball bearings.