Questions about Relays
What's the difference between Rated Carry Current and Max Switching
  Current? Which of these two should I consider as the current limit?


The rated carry current is always the higher figure - basically if you switch a relay on/off at its maximum carry current then you are doing the contacts harm - you should only switch at the max switching current. If that current then grows to the rated carry current that is fine but, before switching the relay off the current has to fall-back to the max switching current.

It won't instantly "fail" if you don't follow the rules but its life will be impaired and it will eventually fail sooner because of this.