My sense is that agape and caritas have come to possess very similar, indeed all but identical, connotations. Etymologically, however, there remains a small, but perhaps significant, difference: the Latin caritas (n.) is cognate with carus (adj.), meaning “dear, highly prized, (even) high-priced”; hence caritas carries the sense of “esteem, affection, (even) desire”. The Greek agape (n.), on the other hand, is cognate with agapao (v.), meaning “to caress, to care for”.
To make use of C. S. Lewis’s distinction in The Four Loves, there would thus seem to be rather more “need-love” in caritas and rather more “gift-love” in agape. Caritas is a love whose object is estimable, if not esteemed, and it is directed toward something regarded as demanding or meriting love. Agape, however, is a love that reflects or embodies the lovingness of the subject more than the loveableness of the object; in fact in its “chemical purity”, if you will, it exists in spite of—even as a function of—the unloveableness of the object. “Herein is love, not that we loved God, but that He loved us, and sent His Son to be the propitiation for our sins” (1 John 4:10).
Mind you, I’m splitting hairs when I say this! After all, when it comes to Saint Paul’s great hymn to love in 1 Corinthians 13, the Vulgate translates the Greek agape with the Latin caritas precisely, and this comes directly into the English of the Authorized Version as “charity”: “And now abideth faith, hope, charity, these three; but the greatest of these is charity” (verse 13). On the other hand, given the etymological facts, it seems to me worth making a subtle distinguo in order to underscore the fact that in the Christian understanding the God who “is love” (1 John 4:8) is “agapic” before He is “charitable”: He is the One who loves the unloveable into a condition of lovability.
You also asked about the Latin imperative miserere. As far as I can tell, there is no difference whatsoever between it and the Greek eleison; each can mean either “have pity” or “have mercy”. But this dual significance itself suggests another important distinction: pity, after all, is compassion for someone who is suffering in a way or to a degree he does not deserve, whereas mercy is clemency or leniency toward someone who is not yet suffering, or not at least fully or adequately, in the way he does deserve. At the risk of another overly subtle distinction, it therefore occurs to me that it might be worth our while to muse upon the following proportion:
Caritas : miserere (or eleison) as “pity” : : Agape : miserere (or eleison) as “mercy”
In other words, the love that loves what it deems to be intrinsically or essentially worthy of its love is naturally inclined to a compassion that pities, whereas the Love that loves because it is intrinsically or essentially Love as such naturally gives rise to a compassion that bestows mercy.
That’s almost certainly more than you needed, or wanted! But I “love” words, and the playful seriousness of etymology, or perhaps better nirukta, I always find irresistible!