大老 means 年高德劭者, 於一領域舉足輕重者, 高官元老, 幫會黨派長老, etc. It is the standard form used in the Classical Chinese literature (e.g., 孟子離婁上:「二老者,天下之大老也」) and 白話 literature of Yuan-Ming-Qing dynasties, and it continues to be so in Taiwan Mandarin.
佬 came from Cantonese, meaning "adult males". Phrases in Taiwan Mandarin that contain "佬" often carry a hint of derogatory undertone (e.g., 闊佬, 鄉巴佬). 大佬 sounds somewhat like 頭目--I mean, there is some potential negativity like English "alpha male", which may have an "animal, dehumanized" sense. 國民黨大老 is a neutral term, free of judgment; 國民黨大佬, on the other hand, may possibly come across as having prejudgment (like 共黨頭目). To be safe, use “大老” not "大佬” in Taiwan.
I have no idea why 在大陆这个词一般写作“大佬”. I'm curious as well (Perhaps borrowed from Hongkong?).