Difference between $\operatorname{Var}(Y)$ and $\operatorname{Var}(Y\mid X)$? What is the difference between $\mathrm{var}(Y)$ and $\mathrm{var}(Y\mid X)$? If $Y = c + \beta X$ and $\operatorname{var}(X)=\sigma^2$, won't both come out to be the same, i.e., $\beta^2\sigma^2$?

Note that we always have $E[f(X)|X] = f(X)$. Loosely speaking, given $X$ there is no randomness in $f(X)$, so we expect the conditional variance to be zero.
Let $f(x) = c+ \beta x$.
\begin{eqnarray}
\operatorname{var} (f(X)|X) &=& E [ (f(X)-E[f(X)|X])^2 | X] \\
&=& E [ (f(X)-f(X))^2 | X] \\
&=& E[0 | X] \\
&=& 0
\end{eqnarray}
On the other hand (note that in this case we have $E[f(X)] = f(EX)$).
\begin{eqnarray}
\operatorname{var} (f(X)) &=& E [ (f(X)-E[f(X)])^2 ] \\
&=& E [ (f(X)-f(EX))^2] \\
&=& E[(\beta(X-EX))^2] \\
&=& \beta ^2 E[(X-EX)^2] \\
&=& \beta^2 \operatorname{var} X \\
&=& \beta^2 \sigma^2
\end{eqnarray}
