I'm not sure what distinction you're making between armistice and surrender in this particular case. Neither Grant nor Lee had the authority to negotiate for their respective governments, only for the armies under their command. However, their respective armies were the primary armies on either side operating in the Eastern theater, and Lee was a man of considerable reputation in the South, so his surrender mostly ended the fighting, and, in general, other Confederate forces were sufficiently demoralized upon hearing news of his surrender to surrender in turn, and Lee's surrender is generally regarded as the effective end of the war.
The specific terms that Grant gave to Lee were:
"In accordance with the substance of my letter to you of the 8th inst., I propose to receive the surrender of the Army of N. Va. on the following terms, to wit: Rolls of all the officers and men to be made in duplicate. One copy to be given to an officer designated by me, the other to be retained by such officer or officers as you may designate. The officers to give their individual paroles not to take up arms against the Government of the United States until properly exchanged, and each company or regimental commander sign a like parole for the men of their commands. The arms, artillery and public property to be parked and stacked, and turned over to the officer appointed by me to receive them. This will not embrace the side-arms of the officers, nor their private horses or baggage. This done, each officer and man will be allowed to return to their homes, not to be disturbed by United States authority so long as they observe their paroles and the laws in force where they may reside."
