Run simple bash script to start applications at login
Solution 1:
Go to System → Preferences → Startup Applications, then click Add. Under Command just enter spotify, instead of the script. There's no need to start a terminal for it.
If you need the output, you can change your script:
#!/bin/bash
echo "start spotify"
spotify > /home/username/spotify.log
and set it as the command, rather than Spotify itself.
Make sure the script is executable by opening the file's properties, going to permissions and setting "Allow executing file as program".