strive vs struggle
The two words have similar meanings, as the OP noted, but strive has the additional meaning of working hard towards a goal. For example:
Susan is striving to achieve her dream of becoming a veterinarian
In the pursuit of her goal, Susan may have to struggle with a course requirement of calculus or she may have to struggle to overcome her aversion to snakes or she may have to struggle for years to earn enough money to support herself and pay tuition.
The difference is further illustrated by these two quotes from Little Women, by Louisa May Alcott:
Money is a needful and precious thing, and when well
used, a noble thing, but I never want you to think it is the first or
only prize to strive for.
But I lost her when I was a little older than you are, and for years had to struggle on alone, for I was too proud to confess my weakness to anyone else.
Strive, from The Free Dictionary
to exert oneself vigorously; try hard.
to make strenuous efforts toward any goal: to strive for success. (Emphasis added)
to contend in opposition, battle, or any conflict; compete.
to struggle vigorously, as in opposition or resistance: to strive against fate.
Struggle, from The Free Dictionary
To be strenuously engaged with a problem, task, or undertaking:
struggled for years before breaking through as an actor.
To have difficulty or make a strenuous effort doing something:
struggled to be polite.
To move or progress with difficulty: struggled up the steep slope