I found myself thinking on this idea for a long time, I really not understand the intuition of the difference between uniform convergence and convergence. But when I thought on the definition deeply, I figured that
If I have convergence of $f_n$ so I know for every $x\in A$ and for every $\epsilon > 0$ there is $N$ such that for every $n>N$ $$|f_n-f|<\epsilon$$
but if I take the $N=\max[{N_i}]$ I get exactly the definition of the uniform convergebce.
Is it correct? what is the motivation to define just convergence, or uniform convergebce?
The problem is that when you wrote $N=max[{N_i}]$, you're silently omitting on which set the index $i$ belongs to. In the case of real maps, you have in fact $i \in \mathbb R$, an infinite set. In that case, $N=max[{N_i}]$ may not be defined.
Some examples
Take for $f_n(x) = 1/n$ a map not depending on $x$. In that case, everything is fine.
Now take $f_n(x) = x^n$ defined on the interval $[0,1]$... I let you define the $N_i$ of your question and figure out that $$N=max[{N_i}]$$ is not defined, or said in another way has to be infinite.