I have a real life situation for you: last year, I developed a wireless sensor system at work. What I describe here is not exactly what happened, but it's not far off.
The product in the end runs off a CR2032 coin cell, and I estimate average idle power consumption to be in the 3uA range, which should give me a couple of years of runtime. However, if I make a mistake in my software, a wrong setting there or an inefficient algorithm here, it's rather easy to elevate this to 40uA, which means the coin cell only lasts a couple of months. The 3uA are for the final product, with the high efficiency parts and without any extra testing circuitry. The dev hardware I have right now is much worse in power consumption, let's say, 200uA.
Still, I need to minimalize power consumption now, part by part. The software has a lot of influence on power consumption. For example, I might disable all external interrupts on the microcontroller and instead have the watchdog wake it up every 10ms to see whether something is happening. Or maybe use a 20ms interval, but only if it results in a noticeable decrease in power consumption. Or disable the watchdog altogether and enable external interrupts. You can only really figure out what's most efficient with testing and measuring.
So I'm looking at differences in the single-digit uA range, on top of 200uA total consumption. Using a cheaper DMM which usually has between 1% and 3% accuracy on the DC current range, you can't quite be confident that what the meter is showing you is really there, and is not owed to temperature differences because the DMM is warming up or something like that. Measured values might be different the next day, which makes you hunt after ghost issues that aren't really there.
If you use more expensive meter like the U1242B you mentioned (heh, and incidentally, I used a U1241B for this task), the specified accuracy leads to an error below 0.5uA for the given application. And its precision (repeatability, i.e. if there is an error, at least the error will be the same the next day) is probably an order of magnitude better than that. It really makes development much easier when you don't have to second-guess your instruments.