Great question. A dictator’s powers werepermitted by the Roman Senate — and according to the Roman constitution, for a limited period of time. In theory, a dictator for life would be as powerful as a king; however, because such powers were traditionally granted by the Senate, there was the significance that they could also be taken away by the Senate.
But with a king (that is, “Rex” in Latin), the office of king was in theory granted not by the Senate at all — which was only an advisory body to a king — but by the gods themselves. No vote by the Senate could even challenge the authority of a king. I suppose that by throwing out the Tarquin kings, the Romans had made a statement that they had misunderstood the will of the gods when they had accepted such terrible rulers.
The other difference is that a king passed the kingship onto an heir, that is, a son. It was far from clear that a “Dictator for Life” would pass that office on to a son. So, one (kingship) was permanent; the other was not.
Ironically, when Augustus, the grand nephew of Julius Caesar, became the first true “Emperor,” he ultimately was more powerful than many a king. But because he kept the outward forms of the Republic (such as the Senate) and avoided the dreaded title “Rex,” many Romans felt for a long time they still lived in a Republic.