Does sequential limits coincide with topology limits? For an example, by Alaoglu's theorem, the unit ball of the dual space is weak* compact in weak* topology. Generally speaking, it is not weak* sequential compact, but if we assume it is, my question is, does the limits of the two sense coincide?

If I understand your question correctly, the answer is this: in any topology, if you have a convergent sequence, then it converges as a net (because given a neighbourhood of the limit, eventually all points are in it).

On the other hand, a convergent net can have infinite subsets (notice that I said "subsets", not "subnets") that have many different accumulation points. To see this, let $\\{a_n\\}_{n\in\mathbb Z}$ be given by $$ a_n=\begin{cases} q_n,&\ \mbox{ if }n<0\\\ 1/(n+1),&\ \mbox{ if }n\geq0 \end{cases} $$ where $\\{q_n\\}$ is an enumeration of $\mathbb Q\cap[0,1]$. Then $a_n\to0$ as a net with the usual order on $\mathbb Z$, while every point in $[0,1]$ is an accumulation point of the **set** $\\{a_0,a_1,a_{-1},a_2,a_{-2},\ldots\\}$