Resting and at rest have slightly different meanings. It would probably be more typical to say After the illness I am resting or resting up (slightly informal) rather than using at rest.
At rest means, as the definition says, not moving. In physics, an object is said to be at rest when its kinetic energy is zero relative to the observer's frame of reference. (This from someone with no more than a basic knowledge of physics, but in other words, it isn't moving.)
It can also have the meaning of "at ease," as in Put your mind at rest, which basically means don't worry.
The term is also a euphemism for persons who are dead, as in "gone to his final resting place."
Resting, on the other hand, has the more specific meaning of not engaging in any physical activity, up to having a light sleep. It also as the slightly broader meaning of taking time off of one's general activities. For example, one might be resting after finishing a difficult challenge at work. An somewhat slangy idiom is to "recharge one's batteries."