Alkalis are water soluble bases. In other words, bases which can be dissolved in water to furnish OH- ions are termed as alkali.
NaOH, KOH, Be (OH)2, Ca (OH)2
In general, people use these two terms interchangeably but there is a basic difference between the two. Alkali are hydroxides of alkali metals and alkaline earth metals. They do not include NH3. So as per this definition, NH3 is not an alkali even though it is a base. Similarly, there’s another definition to it which goes this way: The bases which dissolve in water to furnish OH- are alkalis. This definition also excludes NH3.