The difference between table salt and kosher salt is primarily one of structure. If you pour out some table salt and some kosher salt to compare the two, you will be able to see without much help from a magnifying device that the individual grains of salt are much smaller with table salt than with kosher salt. Indeed, the size of the grains will vary from brand-to-brand of kosher salt as well. There is a series of different salt grain sizes from popcorn salt, the tiniest, through table salt, various kosher salts, various sea salts, rock salts, up to slabs of salt the size of a dinner table or larger.
The variation of the size of the salt grains comes from salt being crystalline in nature. There are several examples of crystals in food, from the obvious sugar and salt to the more subtle butterfat.
A crystal is a fairly regular and rigid pattern that molecules like to follow. When atoms get together, there are quite a few options on how they can combine, even with just a couple of different atoms making up the molecule. Crystals grow because there are regular and rigid, and because they follow simple rules. There's a natural way that they can join, and because it's so stable, anyone else who gets in on that pattern is, by definition, also very stable. Any of the atoms who join on but don't match the pattern aren't stable, and therefore they don't last. This makes room for other atoms who are with the program to join in on the crystalline structure. As long as the conditions are right for more atoms to join in on the crystallization, then the crystal will grow.
Conditions for crystal growth are usually that the atoms are free-flowing, so either very warm or preferably dissolved in a liquid such as water, and that there are plenty of the right kind of atoms to join together on the crystallization fun. So if you have a little salt in a lot of water, you will be much less likely to get a crystal than if you had a lot of salt in just barely enough water. One way people cram enough of the atoms into the water is to heat the water, which causes the water to spread out and make room for more other things to dissolve in it. As long as the water is hot, there will be space for the extra atoms. This means that the solution is supersaturated.
After that, you need to encourage the crystals to grow. One method of encouraging crystals is to reduce the amount of liquid. This is how sea salt, especially the very flaky fleur de sel, is grown. You get some extra salty sea water and slowly allow the salt to evaporate. As the water disappears, the salt has nowhere to go, so much of it starts to form big crystalline flakes. Because the process is so slow and chaotic, the flakes are not as regular of a shape as you would get with something more controlled.
If you want to ensure a certain type of crystal is formed, then you can add a seed crystal. If you have a crystal with the right pattern already, adding it to the supersaturated solution will encourage crystals of that structure to form instead of other potential crystalline structures. Because, as I mentioned earlier, if the atoms join with the winning molecule, they are more likely to win. We like rooting for the underdog, but the chances of the underdog beating the favorite are slim. It's the same with crystals. If there's someone really strong and stable out there, adding together with that is much more likely than some other random crystalline structure just popping up randomly.
It's this joining mentality, incidentally, that makes some steps of candy-making kind of tricky. If you are trying to make a caramel, for example, you are boiling down water that contains sugar. This makes a supersaturated solution, which will want to make crystals if it either loses enough water or if it encounters another crystal. If you let sugar crystals form on the sides of the pan above where the bulk of your sugar water is, you could be in trouble. All it would take is one of those crystals to fall in the solution, and crystals would form everywhere. Then you would not have tasty caramel. You want the sugar water to stay fluid until you add extra liquid and fat in, then you can have nice, chewy caramel.
So, we know that the salt is in different sized crystals, and that is the biggest difference between the table salt and kosher salt. The reason why it's a difference is because of the stability of those crystals. Salt does a lot of things in food, from amplifying flavor to moving water around from one part of a cell membrane to another. Its ability to work depends on how much time it has to do what it's doing. Once salt is completely dissolved in water, it will be carried away and distributed throughout the food, unable to work much of its salty magic. In order to do the most good, it either has to be concentrated on one side of a cell membrane, or it has to be right next to your taste buds, depending on what role it is playing at the time.
In a strong crystalline structure, salt is hard to dissolve. Water has to work around at the edges, picking off the weak sodium chloride molecules one at a time. With a larger, stronger crystal, you need fewer molecules of salt to accomplish the same task, because it doesn't go away as quickly. With the smaller table salt (or even popcorn salt) molecules, you have to use a lot more salt to make a difference in your food.
The size of the crystals is also important for how you use it. If you're weighing your salt for a baking recipe, then you don't have to worry so much what you're using, as it will all probably work itself out before the baking is done. If you are doing a volumetric measurement for a recipe, then you'll want to know what kind of salt they were measuring, because kosher salt takes up more space than table salt, so you might put in too much or not enough, depending.
More subtly is salt that is added to food as needed. I am used to Morton's Kosher Salt, which has a larger structure than Diamond Crystals. If I were to fill up my salt cellar with the Diamond Crystal Kosher Salt and tried cooking with that, I would probably end up over-seasoning everything for a while. I have a certain feel for how much salt is necessary at any given time. I have the same problem if I try to use a spoon instead of a pinch for adjusting seasoning. Any time I try to use a spoon and my judgement for adjusting seasoning, everything is much too salty.
There is one other difference between table and kosher, and that is chemical: most table salt is iodized, which prevents goiter. If you don't eat any processed food, and you only used kosher salt, you'd need to find another source of iodine to prevent goiter. I have never known someone with goiter, so I don't think we're in serious danger just yet, but it's worth keeping in mind.