Is it true that the better the trial function, the better approximation for the energy in variational principle?
TL;DR
No this is not true. Although a &quot;better approximation&quot; might yield a lower energy, this is not always the case.
Full answer
There is a major issue in the formulation of your problem: You want

to relate the difference between an approximation $\phi$ and the exact
ground-state wave function ψ0 with the difference between the
approximate energy $E=\int\phi^* H\phi \mathrm{d}\tau$ and the exact energy

The second part of that statement includes a clear mathematical definition of the target property (energy $E=\int\phi^* H\phi \mathrm{d}\tau$). However, there is no clear definition of what a better approximation of the wave function actually is. How do you want to define or measure it, so it can be expressed in a mathematical way?
There a various options to define such a measure, which would allow comparing two trial wave functions and judge their quality. The most obvious and commonly used one is the total energy. But this is the property you want to compare with, so we need something else. Examples for further options are:

overlap with the exact wave function (if available) $\langle \phi|\psi_0\rangle$
expectation values other than total energy, e.g. $\langle r\rangle$ or $\langle r^2\rangle$
any molecular properties of the chosen model system (e.g. dipole moment, excitation energies, etc.)

The next problem is: the choice of the measure for a good approximation might influence the result. For example the overlap $S_i=\langle \phi_i|\psi_0\rangle$ for two different trial wave functions $\phi_1$ and $\phi_2$ (possibly each with a different variational ansatz) might yield $S_1 &gt; S_2$ (meaning $\phi_1$ would be the better approximation), while the total energy of the second trial wave function might be lower (and therefore close to the true ground state): $E_2 &lt; E_1$.
The different measures on how good of an approximation a given trial wave function is do not correlate, therefore there is no general answer which one is actually better. It depends on the property you want to calculate.
There is a even paper available covering this issue and including a more detailed example. Sadly, it is paywalled although it is primarily intended for educational purposes: https://pubs.acs.org/doi/10.1021/acs.jchemed.8b00959

how can I estimate the difference between the approximate energy and
the exact energy?

In essence, this is the central question quantum chemistry is concerned about. And the field developed a whole lot of different methods to achieve reasonable approximations. Thus, I am afraid there is no simple answer to this question.