I hear a lot of people using the words "cure" and "recovery" interchangeably. Are these two words interchangable in epidemiology? What's the difference between these two in terms of medical terminology? If someone has got recoverd from a disease, does that mean they've been cured of the disease?
This is a semantic question, so there is no correct answer across all contexts; I think you will not receive an answer which will be universal. However, these words are used carefully and technically in literature and scientific communication. You are right in that they are sometimes used rather interchangeably in colloquial conversation.
I'll try illustrate with some simple examples; taking a look in medical dictionaries would give you an idea of how the word has been and ought to be used, though remember that all dictionaries are descriptive, not prescriptive.
Treatment refers to the subjection to some agent or action. Biomedically, the effect of the treatment must be quantifiable and comparable with other treatments, or lacks of treatment, or controls such as placebos. A placebo is also technically a treatment. You subject a patient to a dosage of a drug, or a therapy session, and compare the relative efficacy to other treatments, in an operationalized manner, in order to arrive at a demonstrable conclusion about a treatment.
A cure is a treatment which is more widely understood, agreed or shown to resolve disorders. Most typically, 'cure' is used for treatments that completely resolve disorders; you'd say partial cure if it is incomplete in its action, when you can determine the end-goal very well. This is not used as much in biomedicine because usually no one treatment is responsible for full health; aspirin's ability to lower fevers cannot be said to be a cure for fever, only a good treatment; some fevers cannot be eliminated with aspirin, and it takes the action of the body and the exact state of illness for this to be the case. Also, it is vital to be mindful of the difference between symptom and cause: a fever is a symptom, and cures typically deal with the root (underlying) cause. This is the difference between taking antibiotics for infection, versus taking zinc supplements to shorten the duration of a cold. Treatments may or may not target underlying causes; cures directly work on root causes (I cannot think of counterexamples). Case in point: hepatitis B has no cure, only treatments. A potent anti-venom could be said be a cure for venom in the blood, as its action and effect is rather straightforward.
Recovery is usually a judgment of the extent and duration of an ailment. Recovery and remission are related; one could say that a cancer in remission is a requisite for recovery from cancer. A complete recovery after a common cold, for instance, is when one judges that they are no longer infectious, and one's body is not actively fighting the virus anymore; there are no detectable signs of the illness. You could say you have made a full recovery in such cases; for psychiatric conditions, recovery tends to be partial or intermittent, so you have to be specific and use the word carefully.
Lastly, to answer your question:
If someone has got recovered from a disease, does that mean they've been cured of the disease?
One can recover without a cure or treatment. Your body's natural defenses are never counted as a 'treatment' in epidemiology. Curing is always an action, never an inaction. It would be nonsensical. Recovery is a more abstract concept. You can cure many things without full recovery.