The naming started in NMR and it has become the difference between the following two experiments.
Experiment one: Prepare the qubit in a superposition state (apply a H gate) and vary the wait time and then measure in the superposition basis (apply another H gate). The decay time of this experiment is $T_2^*$. We commonly call this a Ramsey experiment.
Experiment two: Prepare the qubit in a superposition state and apply half the wait time and then apply a pi-pulse (X operator) and then the remainder of the wait time and measure in the superposition basis. The decay time of this experiment is $T_2$. We commonly call this a Hahn Echo.
In the second experiment, the pi-pulse refocuses slow noise which depending on the system can be due to many reasons. There are higher order experiments that refocus the noise better and this is an active research area.
To see this imagine the simplest case where the noise can be explained by a Hamiltonian $H = \Delta |1\rangle\langle 1|$ that is constant and unknown.
In the first experiment:
Step 1. Apply H makes $(|0\rangle+|1\rangle)/\sqrt(2)$
Step 2. Wait makes $(|0\rangle+\exp(-i\Delta t)|1\rangle)/\sqrt{2}$
Step 3. Apply H $[(1+e^{-i\Delta t})|0\rangle+(1-e^{-i\Delta t})|1\rangle]/2$.
The probability of getting outcome 0 is $1/2+\cos(\Delta t)/2$.
In the second experiment:
Step 1. Apply H gives $(|0\rangle+|1\rangle)/\sqrt(2)$
Step 2. Half Wait gives $(|0\rangle+\exp(-i\Delta t/2)|1\rangle)/\sqrt{2}$
Step 3. Apply X gives $(|1\rangle+\exp(-i\Delta t/2)|0\rangle)/\sqrt{2}$.
Step 3. Half Wait gives $\exp(-i\Delta t/2)(|1\rangle+|0\rangle)/\sqrt{2}$.
Step 4. Apply H gives $\exp(-i\Delta t/2)|0\rangle$.
The probability of getting outcome $0$ is always $1$ (there is no decay). So the $T_2$ of this experiment would be infinite.
In the more general case $H = \Delta(t) |1\rangle\langle 1|$ the first experiment gives $\mathrm{Pr}(0) = 1/2+\langle\cos(\int_0^t \Delta(t))\rangle/2$ where we have averaged over different shots (runs of the experiment). I similar but different expression can be derived for the second experiment and I will leave as an exercise what happens but depending on the assumptions you are willing to make about the noise correlations you can simplify this expression in terms of the noise spectrum.
If you want to go check it out on a real experiment you can try the notebook https://github.com/QISKit/qiskit-tutorial/blob/master/reference/qcvv/relaxation_and_decoherence.ipynb but you need an account on the IBM Q experience.