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Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...