For an option with price $C$, the P$\&$L, with respect to changes of the underlying asset price $S$ and volatility $\sigma$, is given by
\begin{align*}
P\&L = \delta \Delta S + \frac{1}{2}\gamma (\Delta S)^2 + \nu \Delta \sigma,
\end{align*}
where $\delta$, $\gamma$, and $\nu$ are respectively the delta, gamma, and vega hedge ratios. Then it is clear the vega P$\&$L has exposure to the change of the implied volatility $\sigma$. Note that, for the gamma P$\&$L,
\begin{align*}
\frac{1}{2}\gamma (\Delta S)^2 = \frac{1}{2}\gamma S^2 \frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2\Delta t,
\end{align*}
where $\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2$ is the realized variance, and $\sqrt{\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2}$ is the realized volatility. To see why $\sqrt{\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2}$ is the realized volatility, we assume that, heuristically,
\begin{align*}
dS_t = S_t\left(r dt + \sigma_{Re} dW_t \right),
\end{align*}
where $\sigma_{Re}$ is the realized volatility and $\{W_t, \, t \ge 0\}$ is a standard Brownian motion. Then
\begin{align*}
\sqrt{\frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2} \approx \sigma_{Re}.
\end{align*}
Consider the delta neutral portfolio $\Pi=C-\frac{\partial C}{\partial S}S$. Assuming that the interest rate and volatility are not change during the small time period $\Delta t$. The P$\&$L of the portfolio is given by
\begin{align*}
P\&L_{\Delta t}^{\Pi} &= \frac{1}{2}\gamma (\Delta S)^2 + \theta \Delta t,
\end{align*}
where $\theta$ is the theta hedge ratio. For small interest rate, which we assume to be zero, $\theta \approx -\frac{1}{2}\gamma S^2 \sigma^2$ and $\gamma = \frac{\nu}{S^2\sigma T}$; see, for example, Black–Scholes model. Then
\begin{align*}
P\&L_{\Delta t}^{\Pi} &\approx \frac{1}{2}\gamma S^2 \frac{1}{\Delta t}\left(\frac{\Delta S}{S}\right)^2\Delta t - \frac{1}{2}\gamma S^2 \sigma^2 \Delta t\\
&\approx \frac{1}{2}\gamma S^2 \sigma_{Re}^2 \Delta t - \frac{1}{2}\gamma S^2 \sigma^2 \Delta t\\
&= \frac{1}{2}\gamma S^2 (\sigma_{Re} + \sigma)(\sigma_{Re} - \sigma) \Delta t\\
&\approx \gamma S^2 \sigma (\sigma_{Re} - \sigma) \Delta t \hspace{1in} (\text{assuming that } \sigma_{Re}\approx \sigma)\\
&=\frac{\nu}{T}(\sigma_{Re} - \sigma) \Delta t.
\end{align*}
The cumulative P$\&$L, over the interval $[0, T]$, is then $\nu (\sigma_{Re} - \sigma)$.