Malphono is right. The differences are generally regular and predictable to the point where even a non-speaker like me can tell them apart by looking for a few distinct things (the vowel alternation between o/a, for instance). They developed into different dialects in the same way that any language does: By being spread over a geographic area large enough to form its own regional standards based on the speech found in particular locations of importance. While the East/West division is important and convenient (given the political demarcation of the Byzantine and Persian worlds and all the acrimony that the two empires had for one another), these are really catch-all terms for bundles of even smaller dialects (or, rather, carry over from the days of Classical Syriac, as its modern daughters have inherited this or that feature depending on where they are spoken), such as that which would be known as Chaldean (nearest I can figure, this is the language of Alqosh and its environs; no modern, disinterested linguist considers “Chaldean” a separate language), Turoyo (the speech of the West Syriacs of Tur 'Abdin in Turkey), or the now-extinct Mlaḥsô, which had been spoken around Diyarbakir, also in Turkey (according to Wiki, the last native speaker of this dialect died in 1998; his children are alive and can speak it, but they are spread in out in Syria, Lebanon, and Germany, and have no one to speak it with, so it is effectively extinct)
