Puff pastry is a very rich dough made with roughly equal parts flour and butter. It is rolled out and folded over itself many times to equally distribute the butter and create many layers of alternating butter and flour (as the French call it, millefeuilles or a thousand leaves, a thousand layers). The end result comes out very light and flakey because when the cold dough hits the oven the butter melts and gives off steam, which makes the dough puff up.
Phyllo dough, on the other hand is a very lean dough, which is mixed, allowed to rest and then stretched until it is paper-thin. I am sure it is mostly done by machine these days but I have seen it done by hand. The dough is placed in the middle of a lightly floured muslin lined table, and then several people position themselves at different parts of the table and start pulling on the outside edge of the dough. As they move around the table they stretch the dough and pull it up in the air much like they are throwing a table cloth up and down.
After the dough is stretched and cut into paper-thin sheets, it is layered. Each layer is brushed with butter before another is put on top. Then it is shaped and baked. All these buttery layers produce a unique crispy crumbly texture.