Difference between Euclidean space and $\mathbb R^3$
Solution 1:
Some books will tell you that they are the same, and some will speak of Euclidean spaces in higher dimensions. It's a matter of convention.
However, the space $\mathbb R^3$, when not assigned an inner product, is only a vector space, so that one cannot speak of angles and distances as one would in Euclidean space. And the space $\mathbb R^3$ has an origin, whereas in Euclidean geometry one does not single out a particular point to play a special role different from the roles of all other points.