I've seen something similar before EWR-HKG, though not the difference between the two monitors. At a guess, you have two different quirks happening simultaneously:

1) In trans-polar routes, the map often freaks out if you're headed straight over the pole. I've seen the East/West line on the map flip back-and-forth again, as it can't decide whether you've gone East or West to get to your destination. (When, in fact, you've actually done neither... due North, then due South!)

2) Sometimes the line shows the projected route (the smooth line), while other times it shows the actual route (the "bumpy" one). I've never figured out the determining factor in whether you get to see estimated or actual route info, nor ever seen an option to change it.

It looks like your flight path was effectively trans-Polar, and one of your maps showed the estimated path going Westbound, while the other showed the actual path, headed Eastbound. Weird!