I think one could interpret "two random variables are perfectly correlated" as "the correlation of the two random variables is $1$ (or maybe $\pm 1$)", and
interpret "the two random variables are perfectly dependent" as "two random variables are comonotonic (or maybe comonotonic or countermonotonic)"
Perfect linear dependence
Let $X$ and $Y$ be two random variables. If $|\mbox{corr}(X,Y)| = 1$, then $Y = \alpha + \beta X$ almost surely for some $\alpha \in \mathbb{R}$ and $\beta \neq 0$, with $\beta > 0$ for positive linear dependence and $\beta < 0$ for negative linear dependence.
It can be shown that $\mbox{corr}(X,Y) = 1$ if and only if $X$ and $Y$ are of the same type, that is there exists constants $a\in \mathbb{R}$ and $b > 0$ such that $Y \stackrel{\mathrm{d}}{=} a + bY$, where $\stackrel{\mathrm{d}}{=}$ denote equallity in distribution, meaning that $X$ and $Y$ have the same distribution up to a location and scale transformation.
Perfect positive dependence
The random variables $X$ and $Y$ are comonotonic if they admit as copula the Fréchet upper bound, that is the copula associated to $X$ and $Y$ is $C(u,v) = \min(u,v)$, which is the strongest type of "positive" dependence.
It can be shown that $X$ and $Y$ are comonotonic if and only if
$$
(X,Y) \stackrel{\mathrm{d}}{=} (h_1(Z), h_2(Z))
$$
for some random variable $Z$ and increasing functions $h_1$ and $h_2$. So, comonotonic random variables are only functions of a single random variable; the conditional density of $Y$ given $X = x$ is a Dirac delta function.
The joint distribution of $X$ and $Y$ is
$$
F(x,y) = C(F_X(x), F_Y(y)) = \min(F_X(x), F_Y(y)) ,
$$
where $F_X$ and $F_Y$ denote the marginal distributions of $X$ and $Y$.
