If you hold 2 yarns together which are far apart on the yard standards numbers (0 - 6), you'll see a big difference, but if you hold a number 3 and 4 next to each other, maybe not so much. The numbers are about how thick the yarn is, and if you use a different yarn thickness than the pattern calls for, you will end up with a very different kind of fabric, and sometimes also size. Even if the numbers are right next to each other and to your eye, the yarn doesn't look very different, a good way to see the difference is by looking at the yardage on the label. If you have 100 grams of each for a number 0 and 1, you might get 800 - 900 yards in the 0 yarn, but only 200 in the 1. They don't look much different in their thickness, but it really is big difference.
When it comes to worsted vs. bulky, as Jan said, if you knit the same garment out of each yarn, you'd end up with a much bigger sweater, or if you still got the same gauge, then a much denser fabric. Also, as I mention before about the yardage-- if the pattern says you needed 10 skeins of worsted yarn and you were supposed to get 130 yds from each skein, if you substituted bulky which got only 90 yds per skein, you'd run out of yarn in the bulky.