Is there a difference between saying "Allahumma salli 'ala Sayyidina Muhammad" and "Allahumma salli 'ala Muhammad"? And does saying "As-salamu alaykum" differ from "As-salamu alaykum wa rahmatullahi wa barakatuh"?

There is no difference between saying "Allahumma salli 'ala Muhammad" (O Allah, send prayers upon Muhammad) and "Allahumma salli 'ala Sayyidina Muhammad" (O Allah, send prayers upon our master Muhammad) in the Tashahhud (definition) (testimony of faith in prayer) in terms of validity; both are sufficient. The difference between the two phrases in offering salutations (taslim (definition)) is that the wording with more letters carries greater reward, based on the hadith (definition) of Imran ibn Husayn concerning the increase in reward with an increase in the words of salutation. If someone greets with "As-salamu 'alaykum" (Peace be upon you), it is permissible to respond with "Wa 'alaykum as-salam" (And upon you be peace), but it is better to add "wa rahmatullah" (and the mercy of Allah). It is obligatory to return the greeting with its like or better, according to the Almighty's saying: {And when you are greeted with a greeting, greet in return with one better than it, or [at least] return it [in a like manner].} (An-Nisa 4:86). It is not permissible to respond with a phrase lesser than what the greeter said.