If we use a slit source [of monochromatic light] in a diffraction grating setup we obtain parallel bands on screen. If instead of slit we use a point source we don't have such bands. Can someone please explain why and how does this happen?
The lenses project the image of the light source on the screen. The grating does not affect the propagation in the z-direction, so in one case you have lines that are offset in the x-direction depending on diffraction order and wavelength in the other case you have points that are offset.
So the grating does the same thing in both cases (in geometrical optics language split each beam into diffraction peaks, where the angle between peaks depends on the wavelength).