Printed dipole vs ordinary (3D) dipole
The diameter and cross-section of the dipole element do have some effect on bandwidth (the thicker the element, the broader the bandwidth) but very little effect on directivity. This is because the cross-sectional dimensions (1 mm) are still small compared to a wavelength at 3 GHz (100 mm).
However, the printed version isn't going to be floating in space, but supported by a PCB dielectric, which will have a significant effect. The dipole length for resonance will be shorter, depending on the dielectric constant of the material, and there will be some decrease in efficiency as well.