From the standpoint of the forum member, both have similar functions, allowing reading, posting, replying and uploading images-sunscribing to threads.
I run off an iPhone 4. Many smartphones run tapatalk, now.
I have used tapatalk for a few months and stability-wise, it has been variable. It is becoming adopted by more people as they learn that a smartphone browser that runs DBs more efficiently and quickly actually exist.
Forum runner is relatively new. As a moderator on Holster Shock Forums, I get more functionality to move, tag, close and sticky the threads. It runs faster and formats of posts are easier to follow since quotes are boxed in like conventional browsers.