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.