Is there any difference in the voltage (potential difference wrt center) and current (rate of flow of electrons) distribution for a half wave dipole fed in the middle (dipole cut in the middle,) or fed with a gamma match (dipole one piece) ?
At the highest level, no. Current is still highest in the middle, and voltage highest at the ends. The radiation pattern is practically identical.
Looking more closely, there must be differences, otherwise it would be indistinguishable from a center-fed dipole. Part of the gamma match can be viewed as a shorted transmission line in parallel with the feedpoint. The currents on this transmission line can then be considered the superposition of the common mode and the differential mode. The common mode radiates as a dipole would. The differential mode does not.
The voltage at the feedpoint of a center-fed 50 ohm dipole driven by 100 watts is 70.7 volts RMS. If you gamma feed that same dipole, the voltage at the center is zero. At that center point in a gamma fed dipole, all of the RF energy existing at that point is in the magnetic field and zero in the electric field. At the tips of the dipole where the current is necessarily zero, all of the RF energy existing at those points is in the electric field.
These points, above, exist on a plane that is perpendicular to the dipole. The MOM (method of moments) equations indicate that the electric field, associated with voltage, on the plane drawn between the two (perfectly balanced) feedline wires at the center point of the balanced dipole will undergo complete destructive interference and therefore be zero at all points on that plane. The conservation of energy principle tells us that if there is zero electric energy at all points on that plane, then all of the existing EM energy has to be in the magnetic field associated with the current. Of course, at the center plane of the dipole, there is a time where the electric field is zero while the instantaneous magnetic field is zero indicating that any existing EM energy is located somewhere other than on that center plane.
On the animation in "How can the voltage at the center of a resonant half wave dipole be zero if the input impedance is 75 ohms?", one can see that there are times when the voltage is zero all along the dipole and times when the current is zero all along the dipole. When the voltage is zero, the E-field is zero and all of the EM energy is in the H-field. When the current is zero, the H-field is zero and all of the EM energy is in the E-field.
Note that, unlike voltage and current where we use the dot product for power, when we are dealing with E and H fields, the power is proportional to the cross product, so either E or H can be zero and power (energy) still exist.
Note that the feedpoint of a center-fed dipole is not a point - it is two points with a space (insulator) in between the two points. Ideally, at the point at the center of the insulator, the electric field is zero just as it is with a gamma feed. Since the voltage is proportional to the electric field, the voltage would also be zero as it is with a gamma feed. If the dipole is fed with coax and no balun at the feedpoint, this ideal situation probably will not exist.
To answer the question: There is negligible difference in the radiation patterns between a properly center-fed dipole and a properly gamma-fed dipole.