Advent of Code: "Not Quite Lisp"
The IDE you linked uses CLISP, which is a bit lenient; when I evaluate
that definition, I immediately get two warnings from SBCL:


; in: DEFUN COUNT-UP-DOWN-CHARACTERS-WITH-DIFFERENCE
;     (SETF COUNT-DOWN 0)
; ==&gt;
;   (SETQ COUNT-DOWN 0)
;
; caught WARNING:
;   undefined variable: COUNT-DOWN

;     (SETF COUNT-UP 0)
; ==&gt;
;   (SETQ COUNT-UP 0)
;
; caught WARNING:
;   undefined variable: COUNT-UP
;
; compilation unit finished
;   Undefined variables:
;     COUNT-DOWN COUNT-UP
;   caught 2 WARNING conditions



That is because count-up and count-down weren't defined anywhere.
It's implementation-defined what happens in this case.

So firstly let's define them with defvar:

(defvar *count-up*)
(defvar *count-down*)
...
(defvar +night-before-xmas+ "...")
...


(Constants usually have + as markers, globals *.  Indentation is
usually a bit different as well, I'm only going to show the
Emacs-formatted code without more explanation though.)

Then again, if you want a functional solution, don't use setf (that
is, assignment) or globals at all.  Instead (and in general) prefer
let.

A few other things to make it more idiomatic (for some value of
idiomatic) are the use of the most strict equality operator possible,
which here would be eql (since you check for exact equality between
characters), and not using if if there's only one case, instead when
or unless would be preferable.

(defun count-up-down-characters-with-difference (input-string up-char down-char)
  "..."
  (let ((count-up 0)
        (count-down 0))
    (loop
      for c across input-string
      do (when (eql c up-char)
           (incf count-up))
         (when (eql c down-char)
           (decf count-down)))
    (list count-up count-down (+ count-up count-down))))


loop also has more grammar to compress it more, but I think this is
fine for now.  Also consider using count instead of a manual loop, i.e.:

(defun count-up-down-characters-with-difference (input-string up-char down-char)
  "..."
  (let ((count-up (count up-char input-string))
        (count-down (- (count down-char input-string))))
    (list count-up count-down (+ count-up count-down))))


Or also (count up-char input-string :test #'char-equal) (see COUNT).