Heat transfer vs calorimetry equation
I'm trying to understand the difference between the equations
Suppose, we have two metal rods at and temperature, and we connect them at the end. I want to know, what do the two above equations tell us, regarding this system.
My understanding is that the first equation tells us, that the two systems will reach an equilibrium temperature, and the heat gained by one rod would be the heat lost by the other. This final temperature would depend upon the relative masses, and heat capacities. However, the first equation tells us, how much heat flows out of the first rod, into the second rod, and the final temperature.
Where does this second equation come from, and what does it tell us ? It seems to me, that the second equation would give us the heat transferred from one rod to the other, in some given time, unlike the first one that gives us the total heat transferred.
Combining the two above equations, we can derive the partial differential equation, called the heat equation. My question is, what does my heat equation tell me ? The first equation tells me, the final temperature of two bodies in contact, provided I know the initial temperature, mass and heal capacities.
What does the heat/diffusion equation tell me then ? Does it tell me, the distribution of heat at some given time ?
Does it mean, if I use the heat equation for our problem with the two rods at different temperatures, I'd get the same final temperature as the equation, after some time, and this temperature would be constant ?