well - this is one issue you've encountered by
not changing. If you are supporting customers who still use access 2003 or earlier, then develop/maintain in 2003. If your customers are using 2007 or later then it's time for you to upgrade as well. If you are supporting both types, you need both versions.
the errors I expect to encounter
I must have updated some 40 mdb systems to accdb over the last 6 or 7 years and hit very few problems with a straight upgrade. The main issue for me is a certain element of making the forms look a bit different so it 'looks like an upgrade' and might include adding some new features such as an email function(email did not exist for my earliest systems, at least not in the way it is now

), plus resisting an inclination to 'improve' the system with all the things I've learned in the succeeding years. Any straightforward system should convert OK and probably will not require any changes.
When you upgrade, a new file is created so you can always fall back on the mdb if you need to.
I heard about being able to mix regular and bold text in a memo field with 2010. Is that feature only available with the new format?
yes - plus, font size, italic, underline, color etc. But there are lots of other standard features such scaling controls to form size, more dynamic reports, etc, new controls and field types (although I tend to avoid using the latter since they are not upsizeable and in my opinion are not really for the professional developer)
It is years since I've developed in mdb but I do remember that at one time, 2003 could not access .xlsx formats - they all had to be converted to .xls before importing/linking or whatever. MS may have put a patch in for this.
ADP has disappeared and security is different so if you make use of these things you will have some work to do, but there are plenty of links out there.
I still have one client who has a mdb back end (with accdb front end). The only reason for this is replication which disappeared in 2007 and the way they are set up requires a major investment on their part to change which for now they are not prepared to finance.