The S-300 uses large missiles (25 feet long) that are cold-launched with gas from their canisters before their rocket engines fire. Its a complex system and it requires that the launch tubes be erected with their tubes on the ground. It takes 30 minutes to set up a S-300 battery with radars.
The Patriot uses a smaller and lighter missile that is launched by firing its rocket engine in the canister. This is a simpler, more reliable, and quicker system but it requires that the launch tubes have a backblast area. So, the launchers are held at an angle when firing to prevent damage to the vehicle. The missiles themselves are very agile and they can engage targets outside the launch direction.
