Troubles ! Mostly just that ! It does troubles.
WARNING: disclosure of spouse mecanisms and events is SPOILER (but i avoided the worst ones)
To be more precise:
Your spouse will do various things other the course of the story/playthrough in the game
requesting help all the time (you can refuse or accept) that will lead to:
disclosure of sigma research to enemy /concurrent nations (meaning loosing precious edge over them)
causes diplomatic problems that can lesser relationship with a foreign nation up tothe point of triggering a war with it (if your diplomacy rating was already low)
taking scientist or even agent (out of your 4 members) out of the game for rather extended period AT THE WORST MOMENT
(and not only can't you choose which agent you have to "assign" to your spouse's leisure and service, but it's usually THE agent you need the most at a most crucial time on an urgent mission of yours... could be coincidence, but after 10 runs of same behavior, i doubt it... game seems to have a way to guess which agent is critical part of your current strategy)
after some point, providing a potential diplomatic small boost overall
after some other point i dont want to spoil, causes a major crisis that also affects/endangers your whole ingame strategy... And be also one of the main trigger for "bad ending" (you however can be quick and avoid this to triggeR/happen or having to solve it at all)
I never dared to try (yet) to refuse helping the spouse and check what a bad relationship with them can cause. My guess is that, while a positive/high relationship grants no benefit, i suspect a bad relationship might causes more noticable drawbacks
The only benefit you can get from the choice of your spouse is that the "agent" you choose as significant other wont be an agent on the game board (cannot be recruited by you or by other nations). So if there is an agent you cannot affort to recruit for your team but are afraid of meeting against your team as an enemy agent of another nation, then you can simply get rid of such troublesome "enemy agent" by being in couple with them.
Hope it answers your question