esqlaw, your skepticism reminds me of how people reacted when the first automotive performance/dyno apps came out for the iPhone in 2008 -- apps like Dynolicious and gMeter. Nobody believed they would work. But within a few weeks, they were being written up in auto magazines, PCWorld, and MacWorld, and have even been on TV. In particular, I remember the host of one show tossing his dedicated test hardware (a gTech or gAnalyst most likely) into the trash can because the iPhone apps did it all and did it better.
The apps were better than any of the dedicated portable units you could buy and mount in a car at that time. Back then, the dedicated units only had one- or two-axis accelerometers, limited sampling rate, and no computation capability. In contrast, the iPhone had a three-axis accelerometer capable of 100Hz sampling, a fairly powerful CPU for computations, and a screen to display numerical results and even graphs. So the apps brought new capabilities to the table and were better in almost every way than the dedicated hardware. They were significantly less expensive too. It's another case where the app on the device replaced the combination of a dedicated measurement unit and a laptop.
As far as a theodolite, its usage comes down to technique and understanding its capabilities and limitations. That goes whether it's a traditional optical unit, or an app implemented on a mobile device. In both cases you are subject to a range of potential errors and must rely on good calibrations. Knowing what I do about the iPhone 4 sensors (as well as the PlayBook's), I see zero difference in using the Theodolite app on a mobile device versus using the disconnected combination of a handheld GPS unit, a compass, an inclinometer, and a traditional optical theodolite. If anything, the app benefits by combining the features together in software. So for instance, the GPS position can tell you about the local magnetic declination, which can be used to adjust the magnetometer to give you proper true bearings. The accelerometer/gyro can be used to tell the magnetometer which way is up to accurately turn magnetic field into a bearing. And so on. The combination of all the sensors in software takes it to the next level, in a way that dedicated hardware devices cannot do.
As far as errors go, it doesn't matter if you have a MEMS magnetometer in a mobile device or a Brunton surveyor's compass if there is magnetic interference in the area. And it doesn't matter if you're using a MEMS gyro/accelerometer or traditional inclinometer to measure angle if you're holding it wrong, or it's out of calibration, etc. Same goes for GPS units -- if you don't have a clear line of sight to the sky, it doesn't matter how expensive your unit is.