What's the difference when smoking in a spherical grill and in a smoker?

No, it won't be different. It is, however, a bit harder.

The issues are mainly:

• It's harder to get the heat away from the food. So you end up cooking with some direct heat. This can be avoided by using a small amount of well soaked, large wood chunks and get them smoking with some normal charcoal. You should also use indirect heat by stacking the chunks on one side of the grill. If you use wood chips, this won't work. They burn too fast and need too much of an ignition source.
• Heat insulation isn't very good. The walls are thin and you have to remove the whole lid to tend your food. The heat dissipation can be a real issue for long low and slow recipes. Smokers either tend to be well insulated (brick or ceramic) or have small doors so you can access your food from the side.
• Adding fuel can be a pain. It's made easier if you have a grill surface that is hinged. If you don't, I'd suggest getting one if you are seriously considering smoking. It makes adding fuel much easier. Of course, adding fuel causes the above heat dissipation issues, so add fuel as infrequently as possible.

That said, I smoked for years on a Webber grill, and it worked just fine. I did, however, upgrade to a Big Green Egg at the first available opportunity. The results are more consistent and the process is much easier. But you'll be fine getting started with the webber.

Assuming you are using wood chips in both cases to generate smoke, the main difference is the distance from the heat source. In a smoker, the smoke from the coals has time to cool before reaching the meat, which cooks the meat indirectly, and at a lower temperature. You can get similar results with the normal grill by using indirect heat (placing the coals on one side and the meat on the other), and further separating the the meat from the direct heat by placing some foil between the meat and the coals.

Here is an interesting article on turning your normal kettle grill into a smoker: http://simplyrecipes.com/recipes/how_to_turn_your_kettle_grill_into_a_smoker/

The results aren't really any different between a Weber kettle and a smoker. What does differ is the process. You will have a more difficult time controlling temperatures in a kettle. While you can effectively maintain low temperatures with the grill, bear in mind that it was not designed for that purpose, so you will have to monitor it pretty carefully. Also, your food will be closer to your fuel -- there is no escaping that -- so you will have to rotate the food to ensure even cooking.

However, the results come out largely the same. I use a Weber Smokey Mountain for most of my barbecue, but if I don't need to go as long, I'm doing less quantity, or I want to go with much higher heat, I'll still set up the kettle.