Boots are safer for riding because their smooth surface prevents them from getting caught in the stirrups. This applies to all boots, not just horse riding boots. This is why beginners are often advised to wear normal rain boots or wellies for their first riding lessons. Riding boots made of leather provide additional support for the leg. This leads to calmer and better control.
Another aspect is appearance: riding is a rather traditional sport. When participating in competitions, horse riding boots are also important for visual reasons - especially in dressage. If you don't want to spend so much on boots, boots made of synthetic material are a good choice. Visually, they are often hardly distinguishable from leather boots. However, they do not support the leg as well because the synthetic material is softer than leather.