After running install.php, everything appears to work, but changes are not saved within Store Configuration.
Incorrect ashoppath or ashopurl settings can cause this trouble.
Versions prior to 3.6.3 - Check the ashoppath and ashopurl settings within the config.inc.php file.
Versions after 3.6.3 - Check the ashoppath and ashopurl settings within the database table 'preferences'.
See also: Finding The AShop Path