Why are storms relatively short on Earth?
The main driving force for storms to form at all are the equator-pole temperature gradients. The main reason  for them to stop is friction between the lower atmosphere and earth's surface. This friction counteracts geostrophic balance (the balance between the pressure gradient and the coriolis force). Longer storms can only sustain themself through latent heat release, which can act as a secondary driver (and is the reason that Hurricanes/Typhoons can last rather long). However, at the latest at landfall this mechanism stops and the storm will decay. On gas-planets, however, there is no surface, therefore friction is much less (only the friction between different layers of the atmosphere, which is much less). Once a storm develops, with a low pressure centre, there is no (or hardly) any friction which counteracts geostrophic balance. This allows the storms to exist much longer. 

As to other non-gas planets:
Mars and Mercury have a very thin atmosphere, therefore, as far as I know, there is no weather as on earth anyway. As for Venus, which has an atmosphere, I do not know.

Note that also on Earth, some weather circulations last much longer than the couple of days that storms last, for example the polar vortex, which lasts the whole winter (and is far away from the earths surface). Especially if you compare the gas-planets, you might also see them as "storms" in the wider sense.

references:

(lack) of atmospheres on Mars and Mercury: https://en.wikipedia.org/wiki/Atmosphere_of_Mars, https://en.wikipedia.org/wiki/Atmosphere_of_Mercury

no surface on Jupiter: https://en.wikipedia.org/wiki/Atmosphere_of_Jupiter

disturbance of geostrophic flow due to friction: https://en.wikipedia.org/wiki/Ageostrophy

formation of storms in general: "An Introduction to Dynamic Meteorology - James R. Holton", Chapter 9, and 11