Frankly speaking, I am afraid there is no exact difference between a sty and a hordeolum. It is the same condition with different medical term. Generally, a sty is also called a hordeolum, which appears red, sore and extruding lump on the edge of eyelid. A sty is usually caused by blocked glands on eyelids from bacterial infection. A sty can be categorized as internal and external hordeolum according to different infection part. There is another kind of infection on eyelid which is usually confused with a sty, chalazion. Chalazion is the enlargement of meibormian gland for it is clogged. Meibormian gland is responsible for oil producing. Once it is blocked, the oil will build up in the gland and enlarge the gland. But it is not caused by bacterial infection. If the internal hordeolum doesn't get healed and drained, it will gradually develop into chalazion. It is less painful than a stye. Both of a sty and a chalazion appear a bump on eyelid, just with a little different location, so it is hard to tell. You can apply some warm saline compress on it to help the oil drain for both. As to a sty, you can use some antibiotics ointment to control infection.
Well, of course, they are different from each other. And generally speaking, eye stye is caused by some bacteria and allergies. So for the treatments, you should make sure your eyes are far from the bacteria, so that they can not be infected. Also, burning eyes will be the symptom, so you can just have heating a piece of garlic, which will work wonders if done right. On the other hand, a hordeolum is a small infection of the glands in the eye, which are located in the eyelids. And then, because of the infection, a red bump on the eyelid which looks like a pimple will occur. For the treatments, you can just try some tea bag, which contains many different objects which do a good job to stye.
Although the 3-5 days of the symptoms of hordeolum and stye look the same, there still exists the big difference. The hordeolun is commonly said as steal stythe eye of a needle. The eyelid glands acute suppurative inflammation is often caused by staphylococcus aureus infection according to the affected gland tissue. The outer stye and internal hordeolum outside stye system may have the eyelash, hair, follicle and sebaceous gland inflammation, belonging to within stye meibomian gland of acute suppurative inflammation. After 3 to 5 days, the hordeolum will be in palpebral conjunctiva surface if it is broken at the surface. However the stye is different which is mainly caused by the eyes infection through the coming invisible bacterium. You need to make the great difference between them.