Babble and ramble are both derogatory when applied to speech or writing; but beyond that they have very little in common.
The primary sense of babble has to do with >sound<: as your dictionary tells you, it characterizes speech as “rapid and continuous”, so much so that no coherent meaning can be discerned. We use babble to speak of babies’ meaningless syllables, of mindless social chatter, of fluent but uninformed speech, and of speech in foreign languages which we do not understand. We also use the term figuratively of rapidly flowing streams: ‘babbling brooks’.
The primary sense of ramble has to do with >direction<. It was used originally of walking or travel: to wander aimlessly, going nowhere in particular. When ramble is used of speech or writing it indicates that a discourse is similarly directionless: it jumps from one point to another, with no apparent connection and no evident point towards which it is driving.
Incidentally, you’re very unlikely to hear or read “He is babbling in his speech.” Since you’re talking about a person, it would be taken for granted that it his speech which is involved; we‘d just say “He’s babbling.”
“He is rambling in his speech” is unlikely, too; more usual expressions are:
“He’s rambling”, to describe what he’s saying now as incoherent
“He rambles”, to describe his customary speaking style as incoherent
“His speech rambled” to describe the lecture or speech he gave on a specific occasion as incoherent
“His speech is/was rambling”, to suggest that his manner of speaking is/was symptomatic of inebriation or disease