For example, when my prof defined pointwise convergence and uniform convergence he said something in the line of
pointwise convergence: for every $x$ in set $K$, the sequence $f_n(x)$ converges to $f(x)$
uniform convergence: the sequence $f_n(x)$ converges to $f(x)$ for all $x$ in set $K$
Now these sounds Exactly the same to me! What is the main difference and can anything be worded better?
In fact you're right, they're the same thing. The problem is that the definition you gave of uniform convergence is wrong. I don't know how deep to go then I'll explain everything. When you say that $f_n(x)$ converges to $f(x)$ you are saying that the real succession of the $f_n(x)$ terms converges to the real value $f(x)$, or $|f_n(x)-f(x)| \rightarrow 0$ for $n \rightarrow \infty$. Here you are considering $x$ as a fixed point and you are "moving" n and that is why it's called pointwise convergence.
Instead you say that the succession of functions $(f_n)$ converges uniformly to $f$ if the uniform norm of their difference tends to zero for $n \rightarrow \infty$. You consider then a difference between functions and not adifference between real values as you did before in the pointwise convergence. You here have to show that $$ ||f_n -f || \rightarrow 0 $$ for $n \rightarrow \infty$. Notice that I didn't write $||f_n(x) -f(x) ||$ but $||f_n -f ||$ because I'm treating the functions themselves and not their value for a certain x. (Hope you know how the uniform norm is defined).