Strictly speaking, the "touch down" is when the plane's wheels touch the ground, and (as Jason Bassford says) is only one part of the overall "landing". However, in practice, these are synonymous and you can use either one in everyday speech:

I should get to the airport. My wife's plane is touching down/landing in thirty minutes and she wants me to pick her up.

In some cases, "touch down" can sound slightly more dramatic. Note that it is typically used with spacecraft as well as aircraft:

Engineers at NASA's Jet Propulsion Laboratory in Pasadena, California, which leads the mission, are preparing for the spacecraft to enter the Martian atmosphere, descend with a parachute and retrorockets, and touch down tomorrow at around noon PST.