Sorry OronD but I think your answer is only half correct. Eilat airport, located in the middle of the Eilat resort is for internal flights only. International flights to Eilat go through Ovda airport which used to be a military only airport but for more years now it has also served as a commerical airport and is a 45 minute drive from the resort of Eilat.
Until recently ElAl were only able to run internal flights from Tel Aviv to Eilat through Ovda but now they will also be running internal and connection flights through Eilat airport.