Using Applescript to append user input to existing note in Notes App
This question explains how to modify text on Notes, so here's how that would be implemented in your code:

set noteName to &quot;JAT&quot;
set myDate to date string of (current date)
set myTime to time string of (current date)
set myStamp to myDate &amp; &quot; &quot; &amp; myTime


set JAT to text returned of (display dialog &quot;What's on your mind, Weston?&quot; default answer &quot; &quot; with icon note buttons {&quot;Cancel&quot;, &quot;Continue&quot;} default button &quot;Continue&quot;)

set entry to return &amp; &quot;&gt;&quot; &amp; myStamp &amp; &quot; | &quot; &amp; JAT

tell application &quot;Notes&quot;
    tell account &quot;iCloud&quot;
        set body of note noteName to (get body of note noteName &amp; entry)
    end tell
end tell

quit


Don't forget to create the note first, though.