I think Lucia's explanation is very good in choosing the correct
English terms.
Let me offer some more examples:
/accorgersi/ is a quick process, that occasionally might be key is putting
us out of an immediate danger.
/rendersi conto/ is the result of a mental process that might proceed in the back of
our mind for quite a long time, and more or less suddenly our conscious mind is shown
the results from the inner workings of our not-so-conscious mind.
Not all the details of this process can be easily explained following strict logic,
due to the fact that something is related to emotions or to the comparison of many,
many facts and experiences.
/capire/ is (usually, but there are other usage contexts) related to the realm of logic,
mathematics, Physics, and so on; once you 'hai capito' something you could describe every aspect of it and any relations of its parts, be it a theorem, the quantum properties of light, an engine or a molecular reaction.
Examples:
e1. Non ti sei accorto di quella macchina? Per fortuna hai sentito il mio grido!
e2. Solo adesso mi rendo conto che ho scelto il momento sbagliato per venirti a trovare. Scusami, ci vedremo in un altro momento.
(There is a very nice song by Lucio Battisti that describes this situation: "Fiori di Rosa, Fiori di Pesco",
https://www.youtube.com/watch?v=6XYoJaDi_qo )
e3. Il fisico Arthur Eddington era molto famoso nella Inghilterra degli anni '20. Alcuni dicevano
che fosse l'unico ad avere pienamente capito la teoria della relativita` generale di Einstein.
e4. Spero che tu abbia capito i sentimenti (feelings) che ho per te.
In (e4) /capire/ is not related to logic, but to feelings. As the two contexts are so distant,
we may enjoy the luxury of using the same verb without causing confusion in the meanings.