Out-Of-Universe / Production Answer Edit
In the real world it was because the original series had a low budget.
In-Universe Answer Edit
During Star Trek: Enterprise's fourth season, the difference in Klingon appearance was finally spoken of & a reason was provided, you should review the episodes, Affliction and Divergence for specifics. In universe, it was because the Klingons tried to use genetically engineered, human embryos to give themselves increased strength and intelligence. A side effect of this was that the Klingons had a more human appearance. Unfortunately, the genetically engineered Klingons died after a few days. One of the new genetically engineered Klingons had a disease which changed the disease, making the genetic enhancements contagious. Eventually most Klingons had the disease.
Dr. Phlox found a way to cure the disease, so that it was not lethal, but a side effect was that the Klingons did not have increased strength or intelligence, however, they still had flat foreheads. It is not known how they got their cranial ridges back after The Original Series.