Both anise (Pimpinella anisum) and fennel (Foeniculum vulgare) are native to the Mediterranean region. They are both umbelliferous herbs in the family Apiaceae. Anise is an annual, dying after one season, while fennel is perennial. I've never grown anise, so can't tell you if it self-seeds as readily as fennel, though many herbs in that family do. Fennel is kind of a nuisance, actually; once you let it go to seed it can be hard to get rid of. Though it can also work well as a landscape element.
|Monarda with fennel as backdrop, in the demo garden|
The type of fennel most of us are most familiar with is Florence fennel or finocchio, a variety with a swollen stem called a bulb (but it grows above, not under the ground). Bulbing fennel can be eaten raw in salads or cooked either alone or in mixed dishes to add its unique light flavor. I like to use it instead of celery in bread-cube stuffings. The Giant supermarket I usually buy it at calls it anise - you now know that's inaccurate, but since it's in the computer, good luck getting them to change their minds. At least when the cashier gives it the "what is this strange vegetable?" look you can say "Anise. But it's really fennel."
I've never had great luck growing Florence fennel, though I have managed to get small bulbs from it. This variety is technically a perennial as well, but since you have to harvest the whole plant it's grown as an annual.
Fennel or anise seeds are great for freshening the breath. I like chewing on green fennel seeds as well as the dry brown ones. And then I save the brown ones for my other Scandinavian bread recipes that call for fennel seeds. I think when I run out of the anise seeds in the little jar I will just substitute fennel seeds where anise is called for! Either way, a bright pleasant taste.