Much speculation here.
The Strait of Gibralter is only 14 km wide, which means that on a calm day one could paddle from one side to the other in a crude dugout, or even just a raft. Having Neanderthal musculature would be a help.
If Neanderthals were smart enough to make sails of animal hides, the crossing could be done in a few hours, with favorable winds. They would not even need to known much about seamanship.
Certain Celtic people traveled in coracles, generally round or almost round boats of skins stretched over a wood framework: crude, improbably shaped for a watercraft, but apparently serviceable. Similar boats have been made of reeds, and designs are found in many cultures.
“The Strait of Gibraltar is only 14 km wide, which means that on a calm day one could paddle from one side to the other in a crude dugout, or even just a raft.”
I crossed the Mediterranean twice. It was out of Spain to Morocco and back from Morocco back to Spain. The weather was great both trips, but the sea was bumpy and bad. There is no way these folks would have known before sailing if the sea was awful or not. (No satellites at that time. :) )
(And no, I did not get seasick, but several folks did.)
P.S.
I know you are talking about the Straits, but there can be no difference in the madness of the sea. Difference is, as you said, is about 14 km from one end to the other. But here, one is paddling like crazy.
The distance between Spain and Morocco does take a little sailing time…
The tidal currents are a real problem, though.