Generally in Common lisp we have EQ, EQL, EQUAL, EQUALP -> from the most specific to the most general. The most specific has the shortest name: https://www.cs.cmu.edu/Groups/AI/html/cltl/clm/node74.html
EQL is actually often used, since it is also the default equality test in Common Lisp. For example the named TEST argument has the default EQL. That information is slightly hidden. Example: (find 1 '(2 3 2 1 3)) is the same as (find 1 '(2 3 2 1 3) :test 'eql) , since EQL is the default test. One can pass a different equality test predicate, if needed.
Some background about naming and equality operators in Lisp: http://www.nhplace.com/kent/PS/EQUAL.html
Other languages for example use some of =, ==, ===, is, ... what may the difference be?
There were many different naming conventions in Lisp over the decades. One was, as mentioned above, the more specific/primitive operators have shorter names. Later (70s/80s, Lisp got a lot more operators and at some point there were operators with long names introduced. This was not liked by everyone, since one needed to type longer names - completion was often new then (or slow).
For example CLOS was added end 80s. Thus it uses names like UPDATE-INSTANCE-FOR-DIFFERENT-CLASS, instead of UPDINSTC ;-) .
Generally, a good Lisp IDE should make getting the documentation (and source code) for any operator easy.