Both English words derive from the same Latin root, and refers to any ceremonial event in which participants go through a prescribed or generally understood order of acts in order to commemorate a past and/or symbolic act as a way of reflect corporately on its significance or to mark a phase in the life of the participant/s (these latter ones are often also referred to as 'rites of passage'). Rites and rituals may be religious or not. Birthdays is an example of the latter.
Important Christian rites include the Eucharist (or Lord's Supper/Holy Communion), baptism, and wedding.
In popular usage, however, the word "ritual" is often used to mean an act without meaning, as when we say that "he is just going through the ritual." This is an abusive and wrong way to use a mentally robust word. A rite is a rite precisely because it is full of meaning. Christians, most of all, should not participate in such loose and careless use of words; it is a clear case of "casing pearls to the swines".
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