The ameloblasts are ectodermally derived cells; in the process of tooth formation, they differentiate as a nice neat row of cells around the future "outside" of the tooth.
The embryonic origins of the odontoblasts, the cells which make the dentin, is a matter of some debate, but they are resident in the site of the future pulp cavity, which is generally held to be embryonic mesenchyme. Hence, they are probably derived from mesoderm.
The odontoblasts bear some similarities to the ameloblasts, in that the both cell types are tall and columnar in shape. They lay down precursor materials—pre-dentin and pre-enamel and as they do so, "back away" from it. The precursor material is then calcified as a new layer of precursor is formed next to it. In the course of tooth formation, differential deposition of dentin and enamel creates the proper shape of the tooth.
In brachydont teeth, once the tooth erupts above the gumline, the inductive stimulus the ameloblasts need is lost, and they die off. Thus enamel can't be made anew and injuries to it are permanent. The odontoblasts making dentin, however, live in a nice protected environment and continue their activity throughout life. Thus as the animal ages the pulp cavity grows ever smaller.
This specimen comes from a mouse fetus, and it's early enough in fetal development that no actual calcification of the enamel has yet occurred, hence it's labeled "pre-enamel." Similarly, there is no "dentin" yet, just pre-dentin. The odontoblasts that made it can't be seen in this image, but see below.
The ameloblasts require an inductive stimulus from the embryonic connective tissue just at the top of this field. In brachydont teeth, when the tooth erupts, this material dies, and the ameloblasts lose the stimulus. Hence they die, too. Once a brachydont tooth has erupted no more enamel can be formed, and damage to it can't be repaired except by artificial means. The dentist fills chips and corrosion holes with man-made materials. This is done mainly to protect the dentin, which is much softer than the enamel, and much more subject to attack by bacterial action.
Below the odontoblasts in this picture you'll see the pulp cavity. The pulp cavity contains blood vessels, nerve fibers, and reserves of undifferentiated embryonic mesenchymal material.
The walls of the pulp cavity are made of dentin, overlain by enamel. If the enamel is damaged and not repaired, the dentin is subject to erosion. Eventually if it's left unfixed, the dentin will be perforated and the pulp cavity exposed to infection. This will result in serious problems; in addition to almost certain loss of the tooth, a really bad infection can spread to the surrounding jaw bone.
Mouse tooth; H&E stain, 1.5 µm plastic section, 100x, 400x, and 400x