More than likely you will not see much of a difference between the two phone apps as they operate similarly. Neither will be as accurate as the cheapest GPS based Garmin or other brand unit.
<...>
The Smart phone apps do NOT use GPS for their positioning; even though you are lead to believe that is the case. In real simple terms, the phone apps are triangulating location based on cell towers. Accuracy depends on signal strength, distance from towers and the number of towers. It can easily be plus or minus 150 feet. So as you run down the road, the app will do a good job of giving you the location and pace, but an occasion will hiccup will add distance as the app puts you a little out of position for a reading or two.
DISCLAIMER: I am a software engineer and work for RunKeeper
I am not sure which fitness apps your are referring to, indeed there are some which do not really use GPS but most of those which do not are not actually designed for outdoor tracking.
But this thread specifically asked about RunKeeper and MapMyRun... and I felt compelled to correct the gross inaccuracies in this thread.
Both RunKeeper and MapMyRun most certainly do use GPS signals for tracking. With pretty much all late model smart phones the GPS chipsets are every bit as accurate as any you will find in a Garmin watch. Some older phones have much weaker GPS capabilities and may even lack gyro, accelerometer, or magnetometer sensors which can be used in addition to cell tower triangulation to help augment tracking if GPS signal is weak or missing.
Unfortunately the algorithms for interpreting the raw data are far too complicated (lots of heavy math) to go into here, but suffice it to say that there are likely to be differences from time to time in the tracks recorded by different apps or dedicated gps devices... but generally you'll only see differences when the devices need to augment with other sensors because the GPS signal is not accurate enough.
Rarely, you'll even get phone operating system bugs where the phone OS tells us that it has a strong GPS signal when in fact it does not and that is when you can really get some wacky trip maps that you sometimes see people complaining about and blaming the app when it is usually the OS with the bug, but as the smart phones are getting better this happens less and less. I've pretty much never seen this happen anymore with iOS 5.1+ or Android 4.0+
Dedicated GPS devices from Garmin and others will record bad trips too, I usually run with a Garmin watch in addition to my phone and there are many cases where my Garmin will hit "dead spots" for GPS and start reporting incorrect paces, distances, etc... and my phone always struggles a bit in the exact same spots but I often find that the phone does a little better job of getting it right in these cases (depending on the app, some only use GPS and others have algorithms to augment using other sensors).
As far as which app is better, I won't go into that as if you saw my disclaimer you'll probably know which app I prefer
