What is the difference between these two expressions [Shell Script]

A newbie's here. Well, I'm having some trouble understanding the difference between:

let a=$a+$b
let i=$i+1 

and:

a+=$b
i=$(($i + 1))

There's definitely a difference because the second expression ruined my script, giving a trash value.

Here is my script:

the question is [ the script needs to give to the user the ability to put some marks as input until the user give q ( as exit ) or a negative value, in the end, we need to calculate the middle value of the given marks ]

this script is working but if you change let moyenne=$moyenne+$note and let i=$i+1 with moyenne+=$note and i=$(($i + 1)) it will give a trash value for moyenne (the middle value).

#!/bin/bash

note=0
meyenne=0
i=0

until [ "$note" -lt 0 ]; do
  read -p "Entrer la note (appuyez sur q pour quitter): " note
  if [ "$note" = "q" ]; then
    note=-4
    echo "Exit"
  elif [ "$note" -ge 16 ]; then
    echo "Tres Bien"
  elif [ "$note" -ge 14 ]; then
    echo "Bien"
  elif [ "$note" -ge 12 ]; then
    echo "Assez bien"
  elif [ "$note" -ge 10 ]; then
    echo "Moyen"
  elif [ "$note" -ge 0 ]; then
    echo "Insuffi"
  else
    echo "Exit"
  fi
  if [ "$note" -ge 0 ]; then
    let moyenne=$moyenne+$note
    let i=$i+1
  fi
done
moyenne=$(($moyenne / $i))
echo "la moyenne est $moyenne de $i notes"

The relevant section of man bash:

   In  the context where an assignment statement is assigning a value to a
   shell variable or array index, the += operator can be used to append to
   or  add  to  the variable's previous value.  This includes arguments to
   builtin commands such as  declare  that  accept  assignment  statements
   (declaration commands).  When += is applied to a variable for which the
   integer attribute has been set, value is evaluated as an arithmetic ex‐
   pression and added to the variable's current value, which is also eval‐
   uated.  When += is applied to an array variable using compound  assign‐
   ment  (see  Arrays  below), the variable's value is not unset (as it is
   when using =), and new values are appended to the  array  beginning  at
   one  greater  than  the  array's  maximum index (for indexed arrays) or
   added as additional key-value pairs in an associative array.  When  ap‐
   plied  to  a  string-valued variable, value is expanded and appended to
   the variable's value.

Since you haven't set the integer attribute for a, a+=$b will perform string concatenation instead of arithmetic addition:

$ a=1; b=2; a+=$b; echo "$a"
12

whereas

$ unset a b
$ declare -i a=1; b=2; a+=$b; echo "$a"
3

Alternatively, you can force arithmetic evaluation using (( ... ))

$ unset a b
$ a=1; b=2; ((a+=$b)); echo "$a"
3

(note that ((a+=b)) also works; the $ isn't necessary to dereference variables in an arithmetic context).