I am seeing differences is the displayed information for person who has posted on the forum depending on the device we view it on.
For example, on a MacBook Pro everything looks correct. We see the persons display name and their rank (either 'new member', 'member' or 'forum team').
However on a iPad we see all the forums they have permission to post in. See the attached screen shot.
And then on an iPhone we only see the display name and no rank. See the attached screen shot.
Can you help me understand why there is a difference and how I might correct it on mobile devices so it matches the display settings on the computer?
Let's deal with tablets first. There is no reason why a tablet - and this applies to most tablets - can not display exactly the same as a desktop. If you go to the forum admin > themes > mobile tablet theme and ensure that 'mobile theme support' is not checked then tablets and desktops should display identically. You can test that here on this site.
As to phones then because the display is so limited we remove some of the data on purpose. If you want to show more then the templates can be edited and changed to do so.