Yes. Puppies grow very rapidly, thus need adequate calories and nutrients. They have specifically high requirements for proteins, certain amino acids (when compared to adult dogs), and need nutrients not only at specific levels, but also in the right proportions. Calcium and phosphorus are important minerals during growth, and only a specially designed food for puppy can ensure their adequate intake.
Small and toy breed puppies need to be fed puppy dog food until they are 9 months old. Medium breeds need to be fed  puppy dog food until they are 12 months old, large breeds until they are 18-month-old and giant breeds need to be fed puppy dog food until they are 24 months old.