Great question! Yes, size and weight. Fighter aircraft need the kind of radar that allows them to detect and track targets. The majority of military fighter aircraft have attack and reconnaissance radars operating in the 8.5-11 GHz and 13-18 GHz bands with a large number operating in the upper portions of the 8.5-11 GHz band.
Weather radars are different animals which operate on one or more of three groups of frequencies depending on the environment they are analyzing.
- S-band (2700–2900 MHz), in tropical and temperate climate areas, for
example, in areas where hurricanes, tornadoes, large hail and monsoon
or heavy rain are common.
- C-band (mainly 5600-5650 MHz band), in climates where attenuation by
intervening heavy rain or large hail is a very minor issue.
- X-band (9300-9500 MHz), in shorter-range hydrological and
meteorological applications such as urban and mountain valley
"Airborne weather radars, which require greater directivity, operate in 5.2-5.9 GHz and 8.5-11 GHz bands. The choice of these two bands for airborne weather radars indicates a dual trade-off of functional requirements: one is between storm penetration/scattering and the other is storm penetration/equipment size. If there is too much scattering, the radar will not penetrate deeply enough into the storm to see its full extent. However, if too little energy is scattered back to the radar, the storm will not be visible on the radar scope. The larger aircraft use 5.2-5.9 GHz band weather radars, even though they are larger and heavier, because of the better storm penetration capabilities and longer range performance. The majority of smaller aircraft employ lighter weight 8.5-11 GHz band weather radars that provide adequate performance."
Because of these differences, designers would have had to install two radar systems, or a much more complex multimode system. To fighter jets, attack and reconnaissance is much more important to the missions they undertake than analyzing weather.
Technology is constantly improving, though. Many aircraft these days have avionics packages that rely on ground data links to obtain accurate weather depictions. It makes much more sense to do that than to try to jam more hardware into already tightly packed avionics bays. There are better ways to use that weight and space, such as for fuel or ordnance.
Sources: Radars and the Electromagnetic Spectrum Introduction