What is the difference between a space plane and a space shuttle? It seems that they both look similar and also have a similar purpose. But what is the true difference?
An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, propeller, or rocket engine.
All space shuttles ever built are ballistic projectiles on the way up and gliders on the way home, neither of which fits the definition of a plane: going up there is thrust but no aerodynamic lift, while coming back there is aerodynamic lift but no thrust.
Great question! It's like asking what's the difference between a fruit and an apple--an apple is a specific type of fruit. Similarly, the space shuttle(s) were specific examples of spaceplanes. There are a few other space planes that exist and many which have been designed: look up the X-37B, Dream Chaser, or Skylon. The shuttle was just the first spaceplane to actually get built.
There might be a slight terminology subtlety in that a spaceplane is any space-going vehicle which uses wings to generate lift. This could include flying up to the edge of the atmosphere using its wings before entering ballistic orbit. The shuttle didn't do that, and neither does/will the X-37B or Dream Chaser. They launched vertically and use their wings when reentering. However, spaceplanes have been proposed which do "fly" to space, and the Skylon concept is one of those designs.
A spaceplane is any spacecraft which can fly like an airplane or a sailplane while it's in the atmosphere. Every spacecraft that acts like that is a spaceplane.
"The Space Shuttle" is the name of a space launch system used by the United States from 1981 to 2011. The Space Shuttle consisted of two solid rocket boosters, one external tank, and one spaceplane. That spaceplane was officially called "the Space Shuttle orbiter," but a lot of people just called it "the Space Shuttle."
The phrase "Space Shuttle" refers only to that one American design. Even if somebody else designs a similar spacecraft, their spacecraft won't be a Space Shuttle.
A "shuttle" is some mean of transportation moving on a very frequent schedule (like an "airport shuttle", bus). By the original meaning, a shuttle is a part of the weaving machine that makes lots of movement across the full width of this device.
A spacecraft of the completely different construction, envisioned for the frequent flights, could probably be called the "Space Shuttle" anyway.