From what I understand on the Japanese Wiki, "Boost spoils of war" raises the droprate by a percentage. Apparently, Extra-Large raises droprates by 50%, so if items/senki have a 10% droprate, it's increased to 15%.
Luck only affects the bonus rewards chance obtained after a battle. So Spoils of War would be better for farming senki-fodder or evolution materials, while Luck can nab you extra rewards specific for end-of-battle bonuses, such Daruma in the Dojo or even an extra Non-human from the daily events.