What are the differences between missiles with a second rocket motor
What you mean second rocket motor, it having literally 2 rockets like SM-6, Stunner; or Dual Pulse (its 1 rocket motor that reignite later)?
In any case, SM-6 and Stunner uses 2 stages (*AIM-152 had one proposal like this, 2 motors, a booster that would be left behind after it burns all propellant, and a second one on the main missile body) because they're based on ground/sea, so very high air density and its on inertia. The motor and casing itself represent a significant % of the overall missile weight, so when the motor run out of fuel is kinda dead weight (if it doesnt increase the drag, it will give more momentum to the missile), we have a perfect example of this, the AIM-174B is basically a SM-6 without the Mk 72 booster, they removed the booster coz its very heavy and the wider diameter of it probably would make harder to adapt to the F-18s;;; the Dual Pulse ones (PL-15 and probably AIM-260) have one rocket motor, but it has two sections of propellant separated, the first one provides speed to loft and flee from the draggier air, and the second one will reignite in a better enviroment, providing a 'better' efficiency to the acceleration;;; Ramjets like meteor usually have a rocket motor just to provide enough speed to start the Ramjet engine, because they require a higher airspeed to run in their best efficiency; i dont know if theres any launch limitation or booster detatchment restriction to make a "dual stage" Ramjet, but theorically seems a good idea, Meteor is said to have significant longer range than 200km, and its rocket motor will make it reach only mach ~2ish after it runs out, just the minimum to make the ramjet work properly, imagine a bigger booster that could boost it to mach ~4ish, it would not only make its range bigger, but decrease time to target, increase the NEZ even more, and avoid the slowish acceleration to cruise speed (mach ~3,3 for 100km distance), could see this working even for a land base anti-air system (for this last one, we can take the NASAMs example using AIM-120C-7; because of the conventional rocket motor, the C-7 doesnt perform THAT well from the ground, if a Meteor with Booster is put in the same scenarion, the Booster would provide altitude and speed enough to even mimick a regular air launch, so it would have probably around its regular maximum range)
sorry if the text is confuse to read, my english isnt that good, but i tried to make it clear