There are several reasons for this. One of them is air traffic concerns. The parts of the atmosphere that are interesting for weather (namely, the troposhere and lower stratosphere) also happen to be the ones in which nearly all aircraft fly. This is why air traffic control is informed and NOTAMs (Notices to Airmen or, as the U.S. now calls them, Notices to Air Missions) are issued when weather balloon operations will take place. For a normal balloon launch, the balloon will only remain at potentially-hazardous altitudes for a matter of minutes (and only a few minutes at the altitudes with the most air traffic density,) so planning air traffic around the launches isn't too big of a deal. However, if the balloon were staying continuously at altitudes where it was collecting useful information, it would also be continuously a hazard to air traffic.
An additional reason, as other answers have mentioned, is that the point of weather balloon launches is to collect data for all altitudes from the surface up to at least the lower portions of the stratosphere, not just one altitude. So, the balloon's flight consisting primarily of a climb phase followed by termination is a feature, not a bug. The radiosonde attached to the balloon collects data such as ambient temperature, dew point, air pressure, wind magnitude, and wind direction for all altitudes. This information then aids meteorologists in determining all kinds of useful information, such as precipitation amount forecasts, potential for thunderstorm development, what type (if any) of precipitation would be expected at the surface, as well as at higher altitudes (for aviation weather forecasting,) depth of different air masses during frontal passage, winds aloft, jet stream position, direction, and magnitude, etc. The data collected by the radiosonde looks like this:
Recent Radiosonde data from NWS weather balloon launched from Nashville, TN.
Units on the left are air pressure in millibars (with kilometers altitude in red.)
Source: National Weather Service
Additionally, balloons are subject to wind. A lot of wind. Making them stay put in one spot relative to the ground is not so easy, especially in the portions of the atmosphere that are interesting for weather observations, which usually have non-trivial winds aloft, even when the wind is mostly calm at the surface. As seen in the radiosonde data in the picture above, winds aloft were measured up to about 118 knots (136 mph or 219 km/h) in this particular launch. According to the U.S. National Weather Service, even its normal balloon launches can drift as much as 125 miles (200 km) during flight. If you just let the balloons free-float indefinitely, then they won't stay where you want them for long, at least not in the interesting parts of the atmosphere. In order to produce reliable, comparable records, the balloons need to be launched at the specified times from the same positions on the ground. Of course, you can tether balloons to the ground in order to make them stay in a mostly-fixed position, but tethers long enough to reach up to 20 km / 65,000 ft or so are not exactly practical and would also greatly magnify the risk posed to air traffic.