Given the following figure and the following variables and parameters, I have been able to come up with the set of differential equation below the image. My question is what is the assumed difference between latently infected T-cells and actively infected T-cells? In the system of equations the latently infected is $\frac{dL}{dt}$ and the actively infected is $\frac{dE}{dt}$.
Variables
$R(t)$: number of susceptible T-cells (number, $ R(t) \geq$ 0).
$L(t)$: number of latently infected T-cells (number, $L(t) \geq 0).$
$E(t)$: number of actively infected T-cells (number, $E(t) \geq 0).$
$V(t)$: amount of virus (number, $V(t) \geq 0).$
Parameters
$\Gamma$: rate of production of susceptible T-cells (number/time, $\Gamma \geq 0).$
$\tau$: fraction of T-cells susceptible to attack by HIV (unitless, $0 \leq \tau \leq 1).$
$\mu$: removal rate of T-cells (1/time, $\mu\geq0$).
$\beta$: rate of T-cell infection (1/number/time, $\beta>0$).
$p$: fraction of infected T-cells that are latently infected (unitless, $1 \geq p \geq 0)$.
$\alpha$: rate that latent T-cells become activated (1/time, $\alpha \geq 0).$
$\delta$: death rate/removal of actively infected T-cells (1/time, $\delta \geq \mu).$
$\pi$: rate that virus is produced by actively infected T-cells (1/time, $\pi >0$).
$\sigma$: rate of virus removal (1/time, $\sigma >0$).
$$\frac{dR}{dt} = \Gamma \tau - \mu R - \beta VR $$ $$\frac{dL}{dt} = p \beta VR-\mu L - \alpha L$$ $$\frac{dE}{dt} = (1-p)\beta V R+ \alpha L - \delta E - \pi E$$ $$\frac{dV}{dt} = \pi E - \sigma V - \beta V R$$
It's unclear if your question relates to the mathematical or the biological aspects of the problem, but if it's the latter, then a latently infected cell harbours the virus but is not involved in active viral replication or production, whereas an actively infected cell is facilitating the replication and production of new virus particles.
So a susceptible T cell can be infected to become either latently infected or actively infected (that's what those branches with probability $p$ and $1-p$ represent), but only the latter makes new viruses that can infect more susceptible T cells. The latently infected T cells can transform into actively infected T cells (which thereafter start putting out virus particles).