why is Antarctica colder than the Arctic?

Aware that "maritime environments are generally milder/more temperate than continental" and that Antarctica is a continent, whereas the arctic is an ocean, you should see the obvious reason. Do not let thin layer of ice on the surface confuse you, the Arctic ocean is an ocean and the temperature is moderated by the water below the surface. Most of Antarctica is hundreds of miles from the nearest ocean.

Are you sure the Antarctic is colder than the Arctic? And to which extend? Which latitude?

Yes, the coldest registered temperature is the Scott-Amundsen base at the south pole, only followed by a place in Siberia that I forgot the name.

But those are temperature registered at a certain place and not an average for a large area like the entire of the Antarctic continent.

Isolated temperatures can be influenced by a lot of local conditions. For example, did you know that, during a high pressure, at night, the cold air from the mountains sink down in the valleys and it is much colder there?

Yes and sometimes it even causes a wind called a katabatic wind. Sometimes it is so strong and it has been known to destroy Inuit indian villages in the fjords of Greenland when the cold air of the central glacier sinks down to sea level.

1- antarctic is land cover by a mile of ice surrounded by water, while arctic is a few meters of ice on an ocean surrounded by land

2- antarctic had an ocean current circling it that prevents warm water from reaching it

3- antarctic has an average elevation of over a mile of ice