What is the diference between $\lambda x.1$ and $1$?
$\lambda$-contraction is defined as $(\lambda v.M)N \triangleright_\beta M[v/N]$. A $\lambda$-term of the form $(\lambda v.M)N$ is called a redex (= reducible expression). In order to reduce a term, you need a redex -- that's just how $\lambda$-contraction is defined.
The reason that $\lambda x.1$ can not be any further reduced is that it does not contain any redex: It is not and does not contain any terms of the form $(\lambda v.M)N$, so $\lambda$-contraction is just not applicable anywhere. The term thus already is in normal form.
In contrast, $(\lambda x.1)0$ contains a redex, namely itself -- with $v = x$, $M = 1$ and $N = 0$ --, and $\lambda$-contraction yields the normal form $1$.
Obviously $\lambda x.1$ and $1$ are non-equivalent normal forms: One is a function that yields a number, the other one is a number.
$(\lambda x.1)$ is different from $(\lambda x.1)0$ in the same way in which, say, $f: x \mapsto x^2$ is different from $f(2)\ (= 4)$. A function is not the same as the value of the function at an argument, and a function can not be applied to yield a value without an argument present.