Now you can get the ALBUM ARTWORK -- set track_artwork_path to update_artwork(track_album, track_artwork, folder_path)

autoHijackSpotify.scpt

set output_folder to (choose folder with prompt "Please choose an output directory")
set folder_path to POSIX path of output_folder

property file_extension : ".mp3"
-- property file_extension : ".m4a" (* If format is changed to AAC *)

property check_delay : 0.1 (* How often to check for a new track *)
property write_delay : 2 (* How long to wait before updating file name *)
property stop_delay : 1 (* How long to wait before updating final file after playback stops *)

set track_counter to 1

tell application "Spotify"
	if player state is playing then pause
	
	set track_name_x to (name of current track)
	set track_artist_x to (artist of current track)
	set track_album_x to (album of current track)
	set track_artwork_x to (artwork of current track)
	set track_album_artist_x to (album artist of current track)
	--set track_year_x to (year of current track)
end tell

tell application "Audio Hijack Pro"
	activate
	set theSession to my getSession()
	tell theSession
		-- Recording file settings
		set output folder to output_folder
		set output name format to "%tag_comment"
		set comment tag to track_counter
		
		-- Update Track Session data
		set title tag to track_name_x
		set artist tag to track_artist_x
		set album tag to track_album_x
		set album artist tag to track_album_artist_x
		--set year tag to track_year_x
		
		-- Audio format settings
		--set recording format to {encoding:MP3, bit rate:320, sample rate:44100, channels:Stereo, style:VBR}
		--set recording format to {encoding:AAC, bit rate:256, sample rate: 44100, channels:Stereo}
	end tell
	
	if hijacked of theSession is false then start hijacking theSession
	start recording theSession
end tell

set track_counter to (track_counter + 1)

tell application "Spotify"
	-- Start playing Spotify from beginning of current track 
	set player position to 0
	set sound volume to 100
	play
	
	set track_name to (name of current track)
	set track_artist to (artist of current track)
	set track_album to (album of current track)
	set track_artwork to (artwork of current track)
	set track_number to (track number of current track)
	set track_album_artist to (album artist of current track)
	--set track_year to (year of current track)
	
	-- Add delay for resolve 1s recording and stop problem
	delay 1
	
	repeat until player state is not playing
		-- Get new track data
		set track_name_x to (name of current track)
		set track_artist_x to (artist of current track)
		set track_album_x to (album of current track)
		set track_artwork_x to (artwork of current track)
		set track_album_artist_x to (album artist of current track)
		--set track_year_x to (year of current track)
		
		-- On change of track
		if track_name is not equal to (name of current track) then
			my update_artwork(track_album, track_artwork, folder_path)
			
			tell application "Audio Hijack Pro"
				tell theSession
					set comment tag to track_counter
					
					set title tag to track_name_x
					set artist tag to track_artist_x
					set album tag to track_album_x
					set album artist tag to track_album_artist_x
					--set year tag to track_year_x
					
					split recording
				end tell
			end tell
			
			delay write_delay
			
			-- Update the file name from track_counter.mp3 to artist - ... - track.mp3
			my update_filename(track_counter - 1, track_artist, track_name, track_album, track_number, folder_path)
			set track_counter to (track_counter + 1)
			
			-- Get new track data
			set track_name to (name of current track)
			set track_artist to (artist of current track)
			set track_album to (album of current track)
			set track_artwork to (artwork of current track)
			set track_number to (track number of current track)
			set track_album_artist to (album artist of current track)
			--set track_year to (year of current track)
		end if
		delay check_delay
	end repeat
	
	-- Stop recording and edit final file name once playback has stopped
	delay stop_delay
	tell application "Audio Hijack Pro"
		stop recording theSession
		stop hijacking theSession
	end tell
	my update_filename(track_counter - 1, track_artist, track_name, track_album, track_number, folder_path)
end tell

on replace_chars(this_text, search_string, replacement_string)
	set AppleScript's text item delimiters to the search_string
	set the item_list to every text item of this_text
	set AppleScript's text item delimiters to the replacement_string
	set this_text to the item_list as string
	set AppleScript's text item delimiters to ""
	return this_text
end replace_chars

-- Update file name so it can be tagged using Kid3
on update_filename(track_counter, track_artist, track_name, track_album, track_number, folder_path)
	set old_file to ("\"" & folder_path & track_counter & file_extension & "\"")
	set new_file to (replace_chars(track_artist, "\"", "") & " - " & replace_chars(track_name, "\"", "") & file_extension & "\"")
	set temp_final_file to replace_chars(new_file, "/", "-")
	set final_file to ("\"" & folder_path & temp_final_file)
	do shell script ("mv " & old_file & " " & final_file)
end update_filename

-- Update Artwork of current track
on update_artwork(track_album, track_artwork, folder_path)
	set newPath to ((folder_path as text) & (my replace_chars(track_album, ":", "_")) & ".tiff") as text
	try
		set fileRef to (open for access newPath with write permission)
		write track_artwork to fileRef starting at 0
		tell me to close access fileRef
	on error m number n
		log n
		log m
		try
			tell me to close access fileRef
		end try
	end try
	
	return newPath
end update_artwork

-- Set Spotify session in Audio Hijack Pro
on getSession()
	tell application "Audio Hijack Pro"
		set sessionName to "Spotify"
		try
			set theSession to (first item of (every session whose name is sessionName))
			theSession is not null
		on error
			set theSession to (make new application session at end of sessions)
			set name of theSession to sessionName
		end try
	end tell
	return theSession
end getSession

I check it out and hope to make an update :)

I had only now the time to test again your script, following your instructions... and it works!
My fault were using a spotify newer version and adding the script in audiohijack (instead should be used "standalone", after everything)
Only one thing.. is it normal that Album Cover are saved as .tiff images in the same folder as audio files?
Thanks!

Another question: after a while, spotify disconnects and so, the script end to run. Is it because I'm not using premium spotify or simply because after x time it disconnects?

I have the same problem.
But till now i cant solved it i don't finde the problem, i think it comes from spotifiy if the app switch the track.