One difference is that conj accepts any number of arguments to insert into a collection, while cons takes just one:
(conj '(1 2 3) 4 5 6)
; => (6 5 4 1 2 3)
(cons 4 5 6 '(1 2 3))
; => IllegalArgumentException due to wrong arity
Another difference is in the class of the return value:
(class (conj '(1 2 3) 4))
; => clojure.lang.PersistentList
(class (cons 4 '(1 2 3))
; => clojure.lang.Cons
Note that these are not really interchangeable; in particular, clojure.lang.Cons does not implement clojure.lang.Counted, so a count on it is no longer a constant time operation (in this case it would probably reduce to 1 + 3 -- the 1 comes from linear traversal over the first element, the 3 comes from (next (cons 4 '(1 2 3)) being a PersistentList and thus Counted).
The intention behind the names is, I believe, that cons means to cons(truct a seq)1, whereas conj means to conj(oin an item onto a collection). The seq being constructed by cons starts with the element passed as its first argument and has as its next / rest part the thing resulting from the application of seq to the second argument; as displayed above, the whole thing is of class clojure.lang.Cons. In contrast, conj always returns a collection of roughly the same type as the collection passed to it. (Roughly, because a PersistentArrayMap will be turned into a PersistentHashMap as soon as it grows beyond 9 entries.)
1 Traditionally, in the Lisp world, cons cons(tructs a pair), so Clojure departs from the Lisp tradition in having its cons function construct a seq which doesn't have a traditional cdr. The generalised usage of cons to mean "construct a record of some type or other to hold a number of values together" is currently ubiquitous in the study of programming languages and their implementation; that's what's meant when "avoiding consing" is mentioned.