What's the difference between eq, eql, equal and equalp, in Common Lisp?

From Common Lisp: Equality Predicates

(eq x y) is true if and only if x and y are the same identical object.

The eql predicate is true if its arguments are eq, or if they are numbers of the same type with the same value, or if they are character objects that represent the same character.

The equal predicate is true if its arguments are structurally similar (isomorphic) objects. A rough rule of thumb is that two objects are equal if and only if their printed representations are the same.

Two objects are equalp if they are equal; if they are characters and satisfy char-equal, which ignores alphabetic case and certain other attributes of characters; if they are numbers and have the same numerical value, even if they are of different types; or if they have components that are all equalp.

Here are some examples from the same page I linked to above:

(eq 'a 'b) is false. 
(eq 'a 'a) is true. 
(eq 3 3) might be true or false, depending on the implementation. 
(eq 3 3.0) is false. 
(eq 3.0 3.0) might be true or false, depending on the implementation. 
(eq #c(3 -4) #c(3 -4)) 
  might be true or false, depending on the implementation. 
(eq #c(3 -4.0) #c(3 -4)) is false. 
(eq (cons 'a 'b) (cons 'a 'c)) is false. 
(eq (cons 'a 'b) (cons 'a 'b)) is false. 
(eq '(a . b) '(a . b)) might be true or false. 
(progn (setq x (cons 'a 'b)) (eq x x)) is true. 
(progn (setq x '(a . b)) (eq x x)) is true. 
(eq #\A #\A) might be true or false, depending on the implementation. 
(eq "Foo" "Foo") might be true or false. 
(eq "Foo" (copy-seq "Foo")) is false. 
(eq "FOO" "foo") is false.


(eql 'a 'b) is false. 
(eql 'a 'a) is true. 
(eql 3 3) is true. 
(eql 3 3.0) is false. 
(eql 3.0 3.0) is true. 
(eql #c(3 -4) #c(3 -4)) is true. 
(eql #c(3 -4.0) #c(3 -4)) is false. 
(eql (cons 'a 'b) (cons 'a 'c)) is false. 
(eql (cons 'a 'b) (cons 'a 'b)) is false. 
(eql '(a . b) '(a . b)) might be true or false. 
(progn (setq x (cons 'a 'b)) (eql x x)) is true. 
(progn (setq x '(a . b)) (eql x x)) is true. 
(eql #\A #\A) is true. 
(eql "Foo" "Foo") might be true or false. 
(eql "Foo" (copy-seq "Foo")) is false. 
(eql "FOO" "foo") is false.


(equal 'a 'b) is false. 
(equal 'a 'a) is true. 
(equal 3 3) is true. 
(equal 3 3.0) is false. 
(equal 3.0 3.0) is true. 
(equal #c(3 -4) #c(3 -4)) is true. 
(equal #c(3 -4.0) #c(3 -4)) is false. 
(equal (cons 'a 'b) (cons 'a 'c)) is false. 
(equal (cons 'a 'b) (cons 'a 'b)) is true. 
(equal '(a . b) '(a . b)) is true. 
(progn (setq x (cons 'a 'b)) (equal x x)) is true. 
(progn (setq x '(a . b)) (equal x x)) is true. 
(equal #\A #\A) is true. 
(equal "Foo" "Foo") is true. 
(equal "Foo" (copy-seq "Foo")) is true. 
(equal "FOO" "foo") is false.


(equalp 'a 'b) is false. 
(equalp 'a 'a) is true. 
(equalp 3 3) is true. 
(equalp 3 3.0) is true. 
(equalp 3.0 3.0) is true. 
(equalp #c(3 -4) #c(3 -4)) is true. 
(equalp #c(3 -4.0) #c(3 -4)) is true. 
(equalp (cons 'a 'b) (cons 'a 'c)) is false. 
(equalp (cons 'a 'b) (cons 'a 'b)) is true. 
(equalp '(a . b) '(a . b)) is true. 
(progn (setq x (cons 'a 'b)) (equalp x x)) is true. 
(progn (setq x '(a . b)) (equalp x x)) is true. 
(equalp #\A #\A) is true. 
(equalp "Foo" "Foo") is true. 
(equalp "Foo" (copy-seq "Foo")) is true. 
(equalp "FOO" "foo") is true.

Some more notes:

• Most CL functions implicitly use EQL when no test is specified

• See also STRING-EQUAL, = and TREE-EQUAL

• At the core of EQ is usually a pointer comparison

And a rough guide:

To compare against...      Use...

Objects/Structs            EQ

NIL                        EQ (but the function NULL is more concise and probably cheaper)

T                          EQ (or just the value but then you don't care for the type)

Precise numbers            EQL

Floats                     =

Characters                 EQL or CHAR-EQUAL

Lists, Conses, Sequences   EQ (if you want the exact same object)
                           EQUAL (if you just care about elements)

Strings                    EQUAL (case-sensitive), EQUALP (case-insensitive)
                           STRING-EQUAL (if you throw symbols into the mix)

Trees (lists of lists)     TREE-EQUAL (with appropriate :TEST argument)

Note that for efficiency usually EQ >> EQL >> EQUAL >> EQUALP.