For the principle G-bundle $\pi :P\to M$, given a local section $s:U\to P$, it correspond to a local trivialization $$U\times G \to \pi^{-1}(U)\\(x,g) \mapsto s(x)\cdot g$$
where $s(x)\cdot g$ is the action of $g\in G$ on the principle bundle.
I want to understand why this is special for principle bundle which does not hold for general vector bundle.
For a vector bundle $\pi :E\to M$, given a local section $s:U\to P$, it's not sufficient to provide a local trivialization, however if we have a set of local section $\sigma_i$ which form the local frame of the vector bundle, then it can define a local trivalization as :
$$\Phi:U\times \Bbb{R}^n \to \pi^{-1}(U)\\(p,v_1,...,v_n)\mapsto \sum_i v_i\sigma_i(p)$$
Therefore the difference between vector bundle and principle G-bundle is we can use one local section to define a local trivialization for G-bundle, however we need to have n-linear independent local section to define a local trivalization for vector bundle.Is my understanding correct?
Your understanding is not wrong, but the focus on sections is arguably a misdirection. The central issue is, a principal $G$-bundle comes with a $G$-action isomorphic to right multiplication in the fibres. A section of a principal $G$-bundle therefore defines a trivialization just as a choice of origin "promotes" an affine space to a vector space, or a choice of identity element promotes a torsor to a group.
The analogous structure for a vector bundle would be a principal action by the additive group of the fibre. And indeed, such an action exists if and only if the vector bundle is a product, canonically trivialized by its zero section.