What is the difference between ; and ;; in Clojure code comments?

Emacs ; to be used for end-of-line comments and will indent in surprising ways if that is not your intent. ;; does not so I usually use ;;.

Clojure doesn't care - any line is ignored from the ; to EOL.

I believe there is a tradition in CL of using increasing numbers of ; to indicate more important comments/sections.

Check out the official description of the meaning of ; vs ;; in elisp: since the Clojure indenter is basically the same, it will treat them similarly. Basically, use ; if you are writing a long sentence/description "in the margins" that will span multiple lines but should be considered a single entity. Their example is:

(setq base-version-list                 ; there was a base
      (assoc (substring fn 0 start-vn)  ; version to which
             file-version-assoc-list))  ; this looks like
                                        ; a subversion

The indenter will make sure those stay lined up next to each other. If, instead, you want to make several unrelated single-line comments next to each other, use ;;.

(let [x 99 ;; as per ticket #425
      y "test"] ;; remember to test this
  (str x y)) ;; TODO actually write this function

There is no difference as far as the interpreter is concerned. Think of ; ;; ;;; and ;;;; as different heading levels.

Here is my personal use convention:

;;;; Top-of-file level comments, such as a description of the whole file/module/namespace

;;; Documentation for major code sections (i.e. groups of functions) within the file.

;; Documentation for single functions that extends beyond the doc string (e.g. an explanation of the algorithm within the function)

; In-line comments possibly on a single line, and possibly tailing a line of code