Well, there are a few reasons that one might choose the micrometer over a vernier caliper that I can think of. These thoughts are derived from personal experience with the devices.

First:
The resolution of the two calipers can be different.
You will find micrometers in two flavors, one with what is called a vernier scale that allows the user to have a resolution down to 0.0001 inches; the other micrometer type is only down to 0.001 inches and the user can then visually interpolate what the ten-thousands value would be (introducing human error).
As for the vernier calipers, I personally have never seen them with a 0.0001 inch scale...with the exception of a few digital ones...which even then only have an accuracy of 0.001 inches.
So my question is, why even bother with 0.0001" resolution if your accuracy is only 0.001"????? Sorry, just a peeve of mine...

Second:
The construction is different. In a micrometer you have a high precision machined screw which would lend itself to durability.
In a vernier caliper you have gears, which are more subject to wear than a screw.
Point is at some point in time, depending on how nicely the vernier caliper is used and how frequently it is used it will begin to lose its repeatability.

Third:
If the user is using the micrometer correctly they will be using the torque knob to bring the two heads together. By using this knob, you get almost exactly the same pressure applied to the measured sample every time. Which in measuring very thin wires could be highly important, especially becase most wire is copper and copper is very soft and will readily deform!
In using the vernier calipers you have a huge amount of human error that can be introduced into measurements. You can push the measurement head just a bit harder and get a "better" reading. Also, because there is no torque release on the vernier caliper, the pressure applied to the subject piece will likely never be anywhere near the same!

Fourth:
The typical vernier caliper has only one type of measurement head; sure you can buy them with different heads but will you?
The standard "blade style" head that comes on verniers may lend itself to crushing the sample or even severing it if pushed hard enough (see point three).
Micrometers typically have a nice amount of flat surface area and as such reduces the chances for the subject getting smashed. You can also buy kits that have numerous different head shapes that you can install on your micrometer; shapes ranging from points to blades.

All in all, I think that should sum up why the micrometer is superior to the vernier caliper.

Hope that helps!