As others have noted, EPR is a 'more' direct measurement of actual thrust than N1 (although EPR has become a less direct measurement as fan bypass ratios have increased). As an engine guy, the biggest advantage to EPR is that it's temperature independent - at a given pressure altitude EPR will give you pretty much the same thrust regardless of TAT (something you can see in the power setting charts - TO EPR will be pretty much constant below the corner point temperature).
N1 correlates to thrust as a function of air density - and air density varies with temperature and humidity. Temperature can be accounted for with the square root Theta term (aka "root Theta") but humidity is generally an unknown (and can be significant on a hot day), so power setting charts tend to assume worst case for humidity (if it's a hot, dry day you'll get some extra thrust). N1 is also very bad in the event of fan damage (e.g. bird strike) - N1 will go up while thrust is going down.
All that being said, at least from the design engineer standpoint N1 is easier to deal with and has fewer failure modes.
At least to date, pretty much all GE/CFM engines use N1, Pratt and Rolls use EPR. That may change.