Horticulture and agriculture are sometimes used interchangeably, but in reality, there are a lot of fundamental differences between the two disciplines.
Horticulture is best understood as the “science of cultivation”. It is a research-based scientific discipline focusing on the cultivation of crops and how to better their overall performance. Performance of crops is measured by various metrics including disease-resistance, yields, stress tolerances, and nutritional value at harvest.
Most horticultural work is carried out in laboratory settings, and involves a fair bit of biochemistry. The spectrum of research that could fall under a horticulturalist's work is vast, and includes cultivation, plant propagation, breeding of plants, production of crops, plant physiology as well as biochemistry and genetic engineering, among others.
Applications of what is learned in horticulture can be and typically is applied in industrial agricultural settings, so the two disciplines are certainly not entirely siloed. Horticulture involves the study of plants for human consumption, but equally it examines plants that don't make it into our diets.
It is helpful to remember the etymology of the word - coming from the Latin hortus (garden) and cultura (cultivation). It is about how to make the garden grow, so to speak.
Agriculture is the practice of raising crops and animals for human consumption exclusively. It is the traditional idea of "farming" that the public has in their collective mind, and the foundation of the food industry.
Plant agriculture borrows learnings from horticulture to improve disease-resistance, or boost yields, but the goal is not to learn about optimal growing conditions of the plant, it is to produce food for people to eat.
Agriculture also encompasses animal husbandry - the rearing of pigs, cows, sheep, chickens, and other poultry - and aquaculture and fishing practices that give us the seafood we consume.
Agriculture modifies the natural food chain, by redirecting energy in the system. In the natural food web, the first step is the sunlight hitting plants and allowing them to photosynthesise. Then, herbivores eat the plants, and carnivores eat the herbivores.
In plant agriculture, the plants are bathed in sunlight but are then protected from predation by herbivores. Instead, the energy is preserved until harvest and then transferred directly to humans. In this way, we "hacked the system" by writing ourselves out of the food web.
Horticulture is concerned with all plants, both edible and non-edible, whereas agriculture is only concerned with edible plants, but also with animals.
In horticulture, the goal is to understand the optimal growing conditions of a variety of plants. It can be thought of as tending a garden. By contrast, the goal in agriculture is to maximise yields for vast amounts of food production. It is concerned with tending to the field.
Horticulture generally involves small-scale gardening, whereas agriculture typically refers to much larger-scale operations.
Horticulture is a research-based discipline with most work being carried out in the lab. Conversely, agriculture is a very practical job, with most work being done outdoors in the field. That's not to say agriculture doesn't make use of research, it absolutely does. But it isn't the focus of the work.
The relationship between a horticulturalist and agriculturalist is similar to the relationship between a pharmaceutical researcher and a doctor. The researcher spends their days in the lab with theoretical concepts and experiments, whereas the doctor is "in the field" every day talking to patients and making a practical difference in their life.
Of course the doctor couldn't succeed in their role without the work of the researcher, but at the end of the day, their domains are very different.