Why is change in gibbs energy not zero through the gibbs free energy equation?
The change in Gibbs free energy for a closed system is given by:
$$\Delta G = \Delta H - T \Delta S$$
What you are describing is:
$$\Delta H = T \Delta S$$
which is only true for systems at equilibrium, or when $\Delta G = 0$
Why this is true is a little bit difficult to explain. It is essentially a form of the Clausius Inequality at constant temperature and pressure. The Clausius Inequality is really just a statement of the second law of thermodynamics, and it says that at best, for any possible process you can get a zero change in the entropy of the universe, and usually it will be a positive change. From the perspective of the system you can write that as:
$$\Delta S \ge \oint \frac{dQ}{T}$$
which is basically just saying the entropy change of the system is equal to or greater than the sum of the negative entropy changes of the rest of the universe, for each step in a closed cycle.
If you integrate that at constant temperature you get
$$\Delta S \ge \frac{\Delta Q}{T}$$
At constant pressure (and in the absence of non-expansion work) $\Delta Q = \Delta H$, so we can write:
$$\Delta S \ge \frac{\Delta H}{T}$$
...and a little rearranging gives:
$$\Delta H - T \Delta S \leq 0$$
So this says that any process which can occur on its own (without adding work or heat to it) will obey this inequality. That's a pretty useful thing to know, so we let
$$\Delta G = \Delta H - T \Delta S$$
to give us an easier way to talk about it. Processes will happen (on their own) if $\Delta G$ is negative, they won't happen if it's positive, and when they are at equilibrium, $\Delta G$ will be zero.