You already know the synchronic side of things — the rule for how to use it today: the -ent on 3rd-person plural verbs is silent, and on most if not all other words it's pronounced. (Note that it's not only in the present that it's silent, but many 3rd-person plural paradigms, e.g. furent, étaient, seraient.)
On the diachronic side, the history of how it came to be the rule, the explanation has to do with Latin stress patterns.
Latin stress patterns
Latin stress generally fell on either the penultimate syllable, if it was a long vowel, or else the syllable before the penultimate ("antépénultime" selon certaines grammaires) if the penultimate syllable's vowel was short. Gradually, in French, everything but the syllable that was stressed in Latin disappeared, "eroded" away by a lack of emphasis — which is why the last syllable is always stressed in French: the stressed syllable is the one that was strong enough to survive! (Of course, sometimes the vestige of a following syllable is preserved in the form of an e muet and/or silent consonants.)
Thus Auguste Brachet in his Nouvelle grammaire française fondée sur l'histoire de la langue, § 51:
En français la syllabe accentuée est toujours la dernière syllabe [...] le français a conservé l'accent tonique sur la syllabe qu il occupait en latin et il dit finír, aimér labeúr, en élevant la voix sur les mêmes syllabes que les Romains.
Quand, au lieu d'être longue, l'avant-dernière syllabe latine était brève [...] le français a été obligé de resserrer violemment ces mots latins en supprimant toutes les voyelles qui suivaient en latin la syllabe accentuée.
Latin stress in conjugation paradigms
Brachet goes on to apply this system to our friend -ent in § 264, quoted in full here:
Notons aussi que ent (ils chant-ent, finiss-ent, rend-ent) est muet dans ces terminaisons verbales, au lieu d'être sonore et accentué (comme dans souv-ent, auv-ent, arg-ent).
Chant-ons, chant-ez, chant-ent viennent du latin cant-amus, cant-atis, cant-ant. Une fois ces trois finales créées pour la première conjugaison, le français les a employées pour former le pluriel de nos autres conjugaisons, sans recourir pour ces dernières aux formes des conjugaisons latines correspondantes.
Ent est toujours muet dans les terminaisons plurielles des verbes, parce qu'en latin ant (am-ant) était de même inaccuentué, et comme nous l'avons vu § 51, toute voyelle latine inaccentuée, à la fin du mot, devient muette ou disparaît.
[extra hyphens inserted to cope with SE finickiness about italics and bolding]
Seemingly inconsistent treatment of -ent
The question that arises and that you seem to be asking is: okay, sounds good. So why did all the other -ents remain pronounced?
The answer is that they were the stressed syllable in their respective words. According to the rules for Latin stress just described, this means that the ent was not the final syllable in Latin:
convéntum "(un) couvent", subínde "souvent", argéntum "argent"
(Interestingly enough, "auvent" is from a Gaulish root, conformed to the Latin model by analogy.)
Hence the rule applied consistently: these were cut to their last syllable and so were the verbal forms.
You might also wonder why, if we're just cutting words to their final syllables, we end up with -ons and -ez for the nous and vous forms respectively. After all, if a is the last stressed syllable in cantámus and cantátis, shouldn't both those forms end in -a today?
The answer is that with these conjugations in particular, but also with some other words, the final consonants sometimes survived, or an intervening unstressed syllable was deleted or merged instead. There is a good motivation for this process, stated by Camille Chabaneau in Histoire et théorie de la conjugaison française at the end of a longer footnote on these conjugations:
C'est à un besoin analogue, remarquons-le en terminant, celui d'avoir partout des flexions distinctes et sensibles, que les patois obéissent [...]
And it's not only les patois that need to give clear information about the subject of a verb!
This is interesting because, as discussed in another answer, despite hanging on to -ons and -ez at this stage instead of what would have been the identical -a for most inflections, French ultimately gave up its dependence on verbal inflections to distinguish the subject and opted instead for separate subject pronouns: je, tu, il, etc., optional in other Romance languages.
The four irregular forms
As an appendix, there are a few odd cases to look at: vont, font, sont, ont. How did these end up becoming monosyllabic, with the -ant surviving in the form of -ont and becoming stressed?
Probably for the same reason as above: the need to distinguish them. Here are the Latin forms:
Except for sunt, if these forms were cut off at the stressed syllable, they would end in that -a that is so useless for distinguishing the inflection from, say, the 3rd-person singular form. Thus, like nous and vous, those final consonants hang on a little tighter.
(As sumelic shares below via this community wiki answer, in fact, the vowel of the final unstressed syllable also survived in vādunt, habent, and faciunt, which were supposed to have included the post-deletion sequence a_u in popular speech. This diphthong /au/ monophthongized to /o/ in French, the spelling here reflecting that change. The final form of sunt "sont" seems to have arisen by a different vowel shift.)
Aside on Internet resources
Incidentally, an amusing analysis of verb radicals that I happened on in this Wiktionary annexe article, which you may freely ignore since it's utter BS:
Il n’y a aucune raison logique, sinon que les formes conservées jusqu’à aujourd'hui sont toujours celles qui se prononcent le plus facilement.
Whereas here's a surprising gem in a WordReference forum post by a well-informed person!