Good question, can be the source of confusion among a lot of people.
"folding" is a fashion of forge-welding that is designed to help squeeze impurities out of the steel and to sort of "mix" around all the little trace elements and everything. This is because of course billets of steel did not just come out of the ground, they had to acquire their own tamahagane, smelt it into billets of varying amounts of carbon (hagane/kawagane/shingane - high/medium/low carbon content). You would not want to directly weld even amounts of low carbon with medium carbon to get a blade, migration will cause the overall carbon content to be low, but I am assuming you understand that stuff. Anyways...Since the steel is made from a relatively crude starting point, purification is necessary. This is where folding comes into play. Impurities are squeezed out in the folds with the flux and scale. Sometimes welds won't be perfect, and you'll wind up with flaws. That's why you want to avoid overdoing weld cycles. Usually 6-14 "folds" are done to give you an idea. This is of course all done before actually forging the sword, as I'm sure you've gathered.
Lamination tends to refer to the styles of lamination constructions used to create swords. Some people and I have come to believe that this was not really a necessary thing to do, but it was done because you would not always get your kawagane and hagane, and there had to be a way of utilizing the shingane in a sword without it intruding on the ability of the edge. I believe, along with some others, that if tamahagane had a higher level of consistency in carbon levels, they would likely not have taken the laminate constructions.
Let's look at a simple style of lamination like kobuse, which is one of the most common. Core is the shingane, the low carbon stuff that won't harden worth dink. Hagane is jacketed over, forming the edge and sides. There are other styles that are more elaborate, having defined sides, edge, core, and spine...but it went under the principle that they were trying to use all the material they had in an effective manner, and it, in their opinion is much better off bending than breaking. Of course, over time with polishing, that hard jacket of the edge and sides in kobuse will be removed a bit and expose some of the core. This is a standard reason for blades being "tired" if you ever hear that term.
It's not uncommon for swords with cored lamination construction to bend almost like taffy. Some were quite sturdy but of course "great" blades were very outnumbered.
It was generally just a tradeoff some people had to learn to live with, and technique in swordsmanship was paramount to making an extremely effective cut while keeping the sword from twisting or bending. An art of precision that still doesn't fail to amaze me.
I love tamahagane and "folded" blades, but can live without the laminations...I don't really mind them if they're done well though.
Hope I was able to answer your question decently.
Shinryû.