Which are all the differences between uniform convergence and puntual convergence of a sequence of functions {$f_n$} in a compact space?
I know one difference between them, the uniform converge of $f_n$ to $f$ doesn't depend of the point x, whereas puntual convergence does.
Does anyone know about other differences between the convergences?
I guess you were to understand the difference in the requirements of the two modes of convergence of a sequence of functions. Let $X \neq \varnothing$; let $(M,d)$ be a metric space; let $f_{n}: X \to M$ for all integers $n \geq 1$; let $f: X \to M$. Then the sequence $(f_{n})$ is said to converge pointwisely to $f$ iff for every $x \in X$ and every $\varepsilon > 0$ there is some integer $N \geq 1$ such that $n \geq N$ implies $d(f_{n}(x),f(x)) < \varepsilon$. Clearly, pointwise convergence of $f_{n}$ to $f$ requires simply that at every point $x$ of $X$ the sequence $(f_{n}(x))$ of real numbers converges to the real number $f(x)$. From this you can also tell why mathematicians use a word such as "pointwise" or "punctual" to modify such a mode of convergence. On the other hand, the sequence $(f_{n})$ is said to converge to $f$ uniformly iff for every $\varepsilon > 0$ there is some integer $N \geq 1$ such that $n \geq N$ implies $d(f_{n}(x),f(x)) < \varepsilon$ for all $x \in X$. Now uniform convergence of $(f_{n})$ to $f$ instead requires that $f_{n}$ should be arbitrarily close to $f$ if $n$ is large enough.
If $X, M := \mathbb{R}$, then intuitively graphically you can see that uniform convergence of $(f_{n})$ to $f$ requires that the graph of each $f_{n}$ with $n$ large enough should be arbitrarily close to the graph of $f$. However, pointwise convergence of $(f_{n})$ to $f$ does not require that much.