Actually, they sound very different—about as different as French from Italian, and for similar reasons.
In Hebrew, most words are stressed on the last syllable, whereas in Aramaic, it is the penultimate one. This is evident in the oft-recited Kaddish (mourning) prayer, which is perhaps the most commonly recited Aramaic text in Jewish prayers:
Lé-éla min kol birkhata, shirata, vetushbekhata, veneḥemata, damiran bé-alma
which in Hebrew would be:
Lémaalah mikol habrakhot, hashirot, hatishbaḥot, vhaneḥamot shé-amurot ba’olam
Which also illustrates two other distinctions: in Aramaic, the definite article is a suffix (-a), whereas in Hebrew it’s a prefix (ha-)—and Aramaic has no /o/ vowel.
So although the two languages are closely related linguistically (as evident from the roots of the words), the phonology of the two languages is very different.