Caraway Seed, Dill Seed, Celery Seed, Cumin Seed, Coriander Seed, and Aniseed: These all are related products, one-seeded structures of closely aligned plants in the Carrot Family. The Cook and Grocer store them with other herbs and spices, but they challenge the casual definition of both words. I’d guess that many people would think of Dillweed (the leafy part of Dill) and Dill Seed as herbs, while other people would cling to insistence that herbs are stem, leaf, and root, while any flavorful seed and fruit would be considered a spice. My observation is that herb differs from spice based on historical origin. Spices, traditionally, were exotic products that came to Europe from somewhere else (like the Far East) while herbs were traditional ingredients that could be grown locally.
To the Systematist, these various seed are all Umbels, in the Apiaceae (formerly the Umbelliferae). Like Carrot and Queen Anne’s Lace, they produce flowers in umbrella-shaped inflorescences, and generally make paired fruit that break into one-seeded segments.
Thus, to the Carpologist, the structure we call a Dill Seed is a segment of the original fruit that encloses a single mature seed. Since it breaks apart, some call the fruit type a Schizocarp (schizo refers to splitting up), while more formally it is a Polachenarium.