So tad is father and taid grandfather? Is there any difference between taid and tad-cu?
No, just a matter of dialects or simply individual family preference. You will come across both being used quite widely.
Similarly, nain, mam-gu.
taid seems to be more common in the north, tad-cu in the south; similarly with nain versus mam-gu for "grandmother".
And yes, tad/taid is "father/grandfather".