It makes a huge difference. When double-sided printing on an inkjet, I would go no less than a bright white (92 brightness at the minimum) 24 lb. bond.
Laser printers don't have the bleed through inkjets do, so you can get away with 20 lb. bond (which is standard copier paper). I would still seek out a brighter paper, though, 92 brightness mininum. The brighter paper will make the text and graphics pop more and show through the other side less. I'd still do 24 lb. on the laser, though. I think it just feels better.
Of course, depending on how many pages you're doing and how you're going to bind it, you may want to go with a lighter weight paper in the laser, like 20 lb., because it'll make a thinner, lighter book. But, for inkjet, yeah, paper thickness make a huge difference, especially if you're using multi-purpose paper instead of inkjet paper.
My personal choices are:
Inkjet printing: Hammermill ultra-bright white (I think it's like 106 brightness) Inkjet paper, 24 lb.
Laser printing: Hammermill bright white laser paper (92 brightness, 24 lb.) -- though, for character sheets, I find 20 lb. copier/multi-purpose paper works just fine.