"Turn out the light" vs "Turn off the light"

Turn out in turn out the lights is idiomatic.

Turn off is a general verb for turning off almost any device.

The meaning of both is equally the same and there isn't any difference at all, but non-native speakers may not be aware of the idiomatic one and therefore may find it unusual. The less capable ones may not understand the idiom at all. So when not sure how capable your listener is, use turn off.

I will hazard a guess:

There were no more trams in Denmark after 1972 - however my grandmother would tell me to take the tram for another 20 years.

They had petroleum lamps in their summer house, and when you wanted to turn off the light, you turned a brass knob until the wick was low and blew out the flame or turned it further down so the flame went out. Similar with gas lights. That is turning the lights out.

Since then, we flick a switch on and off. Turning the lights off is then taking over.

The expression does not seem to have waned much in popularity though:

Google ngram

As @mplungjan's chart clearly shows, turn the light out is the older form.

I think the reason is simply a matter of increasing domestic electrification through the first half of the last century. Before we had switches, we were much more likely to put the light out

In general, when fire/flame (oil/gas lamps) is extinguished, it goes out, not off.