Why is Gibbs free energy more useful than internal energy?
Thanks for the edit - I see now what you meant.

The short answer for why we "need" Gibbs energy is that the internal energy for a spontaneous process at constant temperature $T$ and pressure $p$ does not necessarily decrease. Internal energy is the thermodynamic potential (wiki link) for constant $S$ and $V$ - this makes sense if you think about it:

If the internal energy decreases, then heat must be released by the system, and/or work must be being done on the surroundings. However, real spontaneous processes happen where heat is absorbed and work is not done on the surroundings - the reason is that if the net entropy increase is greater than the enthalpy decrease, the process will still be spontaneous.

The equation for $\mathrm{d}U$ includes entropy terms, but not in a way that reflects spontaneity - the equation for Gibbs free energy does.

The other reason we "need" Gibbs energy is that it also tells us how much "thermodynamically useful" work we can get out of a process.

Mathematical Explanation:

The two inequalities that you started with only represent spontaneous process under very specific conditions:

$$\mathrm{d}U &lt; 0$$

is true for spontaneous conditions only when $S$ and $V$ are constant. The reason for this is that we define spontaneity as the direction in which total entropy of the universe increases - it is at the root of our concept of "spontaneous" vs. "nonspontaneous."

For example, to arrive at this expression for $\mathrm{d}U$, we start with the Clausius inequality:

$$\mathrm{d}S - \frac{\mathrm{d}q}{T} \geq 0$$

This is just a way of saying that the net change of the entropy of the universe must be greater than or equal to zero for any process, and that the net entropy change of the universe is given by the change for the system ($\mathrm{d}S$) plus the change in surroundings, $-\frac{\mathrm{d}q}{T}$.

To turn this into something useful, we make a restriction: constant volume. This lets us assume no $pV$ work, and we can therefore write:

$$\mathrm{d}S - \frac{\mathrm{d}U}{T} \geq 0$$

and

$$T\,\mathrm{d}S \geq \mathrm{d}U$$

since $\mathrm{d}U = \mathrm{d}q$ at constant $V$.

If we then assume no entropy change for the system, we can also write:

$$\mathrm{d}U \leq 0$$

In other words, the restriction of constant $S$ and $V$ are built-in into the inequality. You can substitute in other thermodynamic variables to get $\mathrm{d}U$ in terms of $T$ and $p$, but it doesn't tell you anything about spontaneity under conditions other than constant $S$ and $V$.

Gibbs energy is useful because the inequality $\mathrm{d}G &lt; 0$ is true for spontaneous processes under constant $T$ and $p$ conditions, which as you mentioned, are easy to achieve in a lab.

The other thermodynamic potentials are derived in similar ways - by holding different variables constant and seeing what happens when change in entropy is greater than zero.