Whether a syllable has a heavy or light rime is often important in whether it will participate in phonological processes, and whether it will receive stress. For example, in Latin, stress is on the penult if it is long in traditional terminology, i.e. heavy (CVV or CVC), otherwise it is on the on the antepenult:
Ca.'tul.lus (the poet) vs. 'ca.tu.lus (young animal)
Of course, languages differ as to whether they treat all types of heavy syllables the same, distinguish between heavy and superheavy syllables, treat geminates the same as other heavy syllables, etc. But is there any good (theoretical) reason why the onset should not count? Does any language treat syllables differently (in some way) depending on whether they have an onset or not, or a simple onset versus a complex onset?