Using three spiders makes clear each is the effect of one spell.
While he could have used the same spider for all the Unforgivable Curses, using three different spiders is a good way to make it clear to the students that the effects caused are solely the result of the spell just cast on it, not a cumulative effect of multiple spells. We know that casting multiple spells at the same object can sometimes change the effect that would happen if only one was cast. For example, when Harry, Ron, and Hermione all tried to disarm Snape (though they were cast all at the same time, not one after the other), instead of just disarming him, they knocked him out and caused a blast - neither of which are typical effects of Expelliarmus.
“Expelliarmus!’ he yelled – except that his wasn’t the only voice that shouted. There was a blast that made the door rattle on its hinges; Snape was lifted off his feet and slammed into the wall, then slid down it to the floor, a trickle of blood oozing from under his hair. He had been knocked out.
Harry looked around. Both Ron and Hermione had tried to disarm Snape at exactly the same moment.”
- Harry Potter and the Prisoner of Azkaban, Chapter 19 (The Servant of Lord Voldemort)
We do know the Cruciatus Curse leaves its victims with residual pain.
“And these words burst from Harry’s mouth; they echoed through the graveyard, and the dream state was lifted as suddenly as though cold water had been thrown over him – back rushed the aches that the Cruciatus Curse had left all over his body – back rushed the realisation of where he was, and what he was facing …”
- Harry Potter and the Goblet of Fire, Chapter 34 (Priori Incantatem)
Using three different spiders makes it clear that the effects the students are witnessing are solely the effect of the spell that was just cast. Even if ‘Moody’ knew it wouldn’t make a difference if he used the same spider, the students had no experience with the effects of Unforgivable Curses, so it made for a clearer demonstration. In addition, it wouldn’t have been particularly difficult or resource-consuming to get two more spiders, so there was really no reason why not to.