An entrance is less specific in meaning, and more general.
Example. The way into a walled city could be an enormous opening or gate.
If I have a fenced garden, the opening in the fence would be called a gate.
Most "gates" refer to an entrance that is outside of a home. However, then can be a gate in a home, and one example is for the enclosure where a baby or child play in a room.
An entrance can be a doorway to a residence or building.
An entrance can also be the opening of a cave in a mountain or the place at which one enters a manmade mine.
A path can be an entrance to a forest.
A particular skill set or ability can offer an entrance into a specific career.
Passing an exam can serve for an entrance into a college or university or school.
One can find that a specific action can serve for an entrance into a certain kind of experience. One can find an entrance into a pleasant state of mind by releasing fears, such as when one follows the adage; "Let Go, and Let God."
Denver, Colorado in the United States has been referred to as
the "Gateway to the Rocky Mountains".