A sounding rocket measures whatever its owner wants it to measure. that is, the people who use sounding rockets simply want to put some kind of sensor at a certain altitude above Earth's surface. Rockets and balloons are some of the means toward that end. They aren't the ends themselves.

A "sounding" rocket simply means one that goes up (often straight up), and comes back down. One hopes it reaches its target altitude, which may be hundreds of miles above the Earth.

The instruments they carry often measure air properties, as you suggest. They may also take photographs of Earth's surface -- commercial satellite surveillance services are not always suitable. High-altitude sounding rockets usually measure the electrical and radiological properties of the near-Earth environment: the condition of the Van Allen belts, the solar particle flux, etc.

Sounding rockets are rockets that don't make it into orbit. They can make it to space. Typically, they're not as big as rockets that make it into orbit, so they're cheaper.

If your experiment needs the vacuum of space or weightlessness, but not for very long, you can use a sounding rocket. If your experiment needs to stay in the air, for example, you're doing hypersonic aerodynamic studies of a lifting body or scramjet, you might want to get it where you want via sounding rocket.