Surface vs infection. Usually a corneal abrasion is just a scrape on the surface layer of the cornea. These tend to heal and are self limited depending on size. A corneal ulcer is secondary to inflammation or infection and causes thinning and destruction of the corneal tissue. An ulcer is more serious and potentially vision threatening.
Progressive. Ulcers, since they are active infections with active inflammation, will progress unless treated. While an abrasion may not heal as quickly without some attention, it will not tend to progress.
Mechanical injury. Corneal abrasions are caused by mechanical injury where the surface epithelial cells on the cornea are scraped off. A corneal ulcer is an infection, usually bacterial, but can be fungal or even viral. Because abrasions can sometimes become infected, antibiotics may be given for both types of problems.
Ulcer is deeper. Corneal abrasion is a scratch that damaged the top layer of cells on the cornea. Often after a doctor checks it, antibiotic drops are used, and the abrasion heals in 1-2 days without a scar. A corneal ulcer or erosion is a deeper injury, beyond the top layer of cells. The eye doctor will see damage to a deeper layer in the cornea. Stronger, more frequent antibiotics are used, and a scar may form.
Ulcer is infected. A simple corneal abrasion is a surface defect of the eye. A corneal ulcer usually implies that there is an active infiltrate (infection) that needs to be treated with antibiotics. Most corneal abrasions can be treated without antibiotics, but they are often used in a preventative way.