LVDT's are very sensitive, but fairly expensive. They use mutual inductance to determine the position of a moving slug. You pass a sine wave through an input, and you measure the envelope of the sine wave out to determine displacement. the PHASE of the output sine wave determines whether you have a positive or negative displacement, so you need phase detection algorithms to provide an output, and the commercial processors tend to be pricey.
Capacitance is defined as
\$ C = \frac{\varepsilon \varepsilon_0A}{d} \$, where A is Area of the overlap of the plates of the cap, and d is the distance between the plates. Thus, the output of a Variable Area device will be linear with area, which you can easily arrange to be linear with displacement. The output of a separation device is proportional to 1/d, which might be VERY sensitive, but it's not linear. You can probably build an amp that linearizes it without too much trouble, but, the world not being a perfect place, you'll likely lose sensitivity.
The circuitry to support either type of capacitive sensor is here. I lean toward some type of 555 timer arrangement as opposed to any type of AC bridge, which I find tricky.