Why do we need "potential” for amortized analysis?
As is unfortunately sometimes the case, Wikipedia is doing a terrible job of explaining what amortized analysis actually is.

The idea of amortized analysis is that while operations may have a bad worst case cost, their average cost could be much lower. Average cost means different things in different circumstances. In amortized analysis, here is what it means:


  Suppose that you have a data structure supporting operations $O_1,\ldots,O_r$. We say that these operations have amortized cost $c_1,\ldots,c_r$ if the cost of the sequence $O_{i_1},\ldots,O_{i_n}$ of operations is at most
  $$
c_{i_1} + \cdots + c_{i_n}.
$$


This definition is a bit simplistic, since sometimes the amortized cost depends on other auxiliary parameters such as the overall number of operations or the number of operations of a specific type.

The potential method is one way of carrying out amortized analysis, that is, of proving that the operations in a certain data structure have some amortized cost. Stated differently, if you want to prove that the amortized cost of $O_1,\ldots,O_r$ is $c_1,\ldots,c_r$, then one of the techniques is the potential method. There are other techniques as well.

How does the potential method work? You define a potential function $\Phi$ on the state of the data structure. The potential function must be always non-negative, and for the initial data structure it must be zero. You then prove that if at state $S_{\text{before}}$ you execute operation $O_i$ which leads to state $S_{\text{after}}$, then the cost $T_{\text{actual}}$ of the operation satisfies
$$
T_{\text{actual}} \leq c_i + \Phi(S_{\text{before}}) - \Phi(S_{\text{after}}).
$$
Given that this holds for all $i$, consider a sequence of operations $O_{i_1},\ldots,O_{i_n}$, and denote the corresponding states of the data structure by $S_0,\ldots,S_n$. The upper bound on $T_{\text{actual}}$ implies that the total cost of these operations is at most
$$
\begin{align*}
\sum_{t=1}^n [c_{i_t} + \Phi(S_{t-1}) - \Phi(S_t)] &amp;= \sum_{t=1}^n c_{i_t} + \sum_{t=1}^n [\Phi(S_{t-1}) - \Phi(S_t)] \\ &amp;=
\sum_{t=1}^n c_{i_t} + \Phi(S_0) - \Phi(S_n) \\ &amp;=
\sum_{t=1}^n c_{i_t} - \Phi(S_n) \\ &amp;\leq
\sum_{t=1}^n c_{i_t},
\end{align*}
$$
which was to be proven.