The answers here are rather flawed. For one, there was plenty of trade in the eastern Mediterranean, and the trade between modern Lebanon and Egypt was very valued. This trade was often varied out on Egyptian ships. More importantly, if you are referring to colonization of the western Mediterranean, that was pretty much limited to the Phoenicians until the middle iron age (the Minoans did not). They are the outliers, so the proper question is why did they do so, not why did the Egyptians not.

As discussed in this other thread, Herodotus reports (with the usual Herodotean caveats implied) that the Egyptians did mount expeditions to explore -- not west into the Mediterranean, but east down the Red Sea into the Indian Ocean and the coast of Africa. If true, this implies that it is not so much a lack of priority regarding trade exploration but rather the direction in which such exploration was most critical.

Many ancient Greeks believed that their civilization and culture came from Egypt. Herodotus states that the Greek Pantheon especially came from Egypt. Argos specifically had a founding myth stating that their founders were from Egypt.

You would find the Egyptian account of WenAmun interesting reading that explains some of the problems they encountered trying to trade around the Mediterranean. Theft/piracy, uncooperative and contemptuous local traders, hostile mobs. Reading between the lines you can sense that Egypt's capacity to project force overseas was limited and other maritime powers knew it. It should be noted that Egypt had to import most of its maritime-grade wood as well. This put it at somewhat of a competitive disadvantage vs Phoenicians.