Allkalis:the substance which is having oH group.if it is dissolved in water ir aqueous solution that can give hydroxyl ions (oH-minus) that type of substances are called as alkalies
base:the substance may or may not be consist oH group but they will act as bases that type of substances are called bases.
BOTH CONSIST OF OH GROUP.BUT WHEN WE DISSOLVE THE ALLKALIS IN WATER OR AQUEOUS SOLUTION IT WILL GIVES US HYDROXYL IONS.
WHEREAS BASE OR DISSOLVED IN WATER OR AQUEOUS SOLUTION IT WON'T GIVE US HYDROXYL IONS. SO WE CAN SAY THAT "ALL ALLKALIS ARE BASES BUT ALL BASES OR NOT ALLKALIS