What is the difference between these 2 equations? Instead of $\Delta$ change it to some general elliptic operator.
Do they have the same results? Which one is used for which?
The relation boils down to time-reversal, replacing $t$ by $-t$. This makes a lot of difference in the equations that model diffusion. The diffusion processes observed in nature are normally not reversible (2nd law of thermodynamics). In parallel to that, the backward heat equation $u_t=-\Delta u$ exhibits peculiar and undesirable features such as loss of regularity and non-existence of solution for generic data. Indeed, you probably know that the heat equation $u_t=\Delta u$ has a strong regularizing effect: for any integrable initial data $u(\cdot,0)$ the solution $u(x,t)$ is $C^\infty$ smooth, and also real-analytic with respect to $x $ for any fixed $t>0$. When the direction of time flow is reversed, this effect plays against you: there cannot be a solution unless you have real-analytic data to begin with. And even then the solution can suddenly blow up and cease to exist. Consider the fundamental solution of the heat equation, and trace it backward in time, from nice Gaussians to $\delta$-function singularity.
If the solution to the first equation i.e. $u_t + \nabla u = f$ is given by $u(x,t;f)$, then the solution to the second equation, $u_t - \nabla u = f$, is given by $u(x,-t;-f)$.
The equation $u_t - \nabla u = f$ is usually called the diffusion equation while the equation $u_t + \nabla u = f$ is usually called the backward diffusion equation.
The diffusion equation is typically used to model, not surprisingly, all diffusive processes like heat conduction for instance. The backward diffusion equation is popular in financial mathematics, and is used to determine the price of various instruments.