Difference between Norm and Distance I'm now studying metric space. Here, I don't understand why definitions of distance and norm in euclidean space are repectively given in my book.

I understand the difference between two concepts when i'm working on non-euclidean space, but is there any even slight difference between these two concepts when it is $\mathbb{R}^k$?
All norms can be used to create a distance function as in $d(x,y) = \|x-y\|$, but not all distance functions have a corresponding norm, even in $\mathbb{R}^k$. For example, a trivial distance that has no equivalent norm is $d(x,x) = 0$, and $d(x,y) = 1$, when $x\neq y$. Another distance on $\mathbb{R}$ that has no equivalent norm is $d(x,y) = | \arctan x - \arctan y|$.

However, in general, when working in $\mathbb{R}^k$ the distance used is one induced by a norm, and 'unusual' distances are typically used to illustrate other mathematical concepts (eg, the $\arctan$ distance gives an example of an incomplete metric space).
The metric space/distance metric is a mathematical construction built up on topological space (definition of convergence, continuity, completeness, separability, connectedness, compactness etc. etc.). The distance operates with point sets and doesn't know the vectors. The Hilbert space is built up on metric space and defines new operation as inner product, vectors and using norms. The metric space itself doesn't know norms and vectors, since as is introduced on Hilbert space. Below is a derivation how to utilize metric space in Hilbert space. 
Reference: Muscat, J. Functional Analysis An Introduction to Metric Spaces, Hilbert Spaces, and Banach Algebras, Springer, 2014.