>>535218691
>and if you do
in this case means
>and if you do summon it
Because the summon comes first in the text. Nothing about the summon depends on the equip happening.
>then
would mean the same thing. Special summon, THEN equip. THEN means it comes after the previous thing. If the previous thing does not happen then how could it come after?
The difference is that "And if you do" means the summon and equip happen at the same time. "Then" means that it happens after. So like, Bottomless Trap Hole could not be used if it said "Then" because the timing would be incorrect from the last thing that happened being the equip.

>I still don't get the difference between the two
Then don't worry about it. They're 99.999% the same and the more important part is that you understand in both cases the second part is conditional on the first but the first is not on the second.