It is defined as a substance that is capable of donating a lone pair of electrons or accepting protons or donating hydroxide ion in an aqueous solution.
Thus, all the nucleophiles are Lewis bases because of their ability to donate lone pair of electrons.
Base or Bronsted base is a unique name of nucleophile when it attacks and forms a bond with a proton .
Example: Sodium ethoxide can act as a base to abstract protons from water and can even act as a nucleophile in reaction with alkyl halides.
However, all nucleophiles can't act as base, for example is a good nucleophile but not a good base.
However, all bases cannot be nucleophiles. Example: Lithium diisopropylamide or LDA because it is very bulky, so an attack over electrophile will lead to a very sterically hindered system. Hence, such an attack is not possible, but instead, it can abstract proton easily and thus act as a good base.