That the Mycenaeans and Minoans knew each other is clear. The Greek tribes moved in from the North, their martial system of chiefdoms in start contrast to the settled agricultural palace centered system of the Minoans. It is only logical, life being what it is, that the martial system would come into conflict with the more settled system and the martial one would win out in the end.
The sad thing in this scenario is that not only did the Mycenaeans conquer the Minoan world, if not in a concerted effort, than through simple force of numbers, but that they also destroyed the Minoan Culture and assimilated parts of it wholesale, leaving almost nothing of a culture that seemed to have a much more enlightened attitude toward gender roles (At least so far as we know) than the conquering Greeks did.
In the Mycenaean world, might made right, and from the palace culture of the Minoans, through their conquest, rose the citadel culture of the Mycenaean Greeks, the greatest, and perhaps best known such citadel cities being Mycenae, from which we have named these people.
Our knowledge of the Citadel Culture that arose in mainland Greece comes first from the epic poems of Homer, Iliad and Odyssey, which tell the tale of a great war between the Greeks, who went by several tribal names, and the large and powerful city of Troy. Secondly, from research into the archaeology of the regions of Greece described by Homer. Lands that were, even to Homer, already legend.
This same archaeological evidence shows us that the first cult that seems to have had any state of permanence in Mycenaean times was the palace Hearth cult. This, being traditionally tied to the Goddess Hestia, always seemed intriguing to me. The great and patriarchal Mycenaeans worshipping Hestia all the way back into pre-history.
But as I have said, there is no reason to believe that just because the Mycenaeans were so strongly patriarchal they did not feel the influence of Goddesses, and the hearth cult seems proof of this.
Since the Minoans were already in the land, it is possible too that the palace Hearth Cult was part of the way they integrated or assimilated the Minoan Culture into their own, and ways in which the Mycenaean attitudes toward women might have made subtle changes that lead to later looser attitudes toward the freedom of women as seen among the Spartans, for example.
But the greatest and most tragic achievement of the Mycenaeans was war, and most importantly, the great Trojan War which eventually ends in the destruction of their civilization.