Most parents introduce one of these and not the other, and the choice is usually accidental. It doesn't have to be.
Where they come from
The Panchatantra was written around the 3rd century BCE for princes who needed wisdom quickly. It's a deliberate, structured tutoring text disguised as fables. Vishnu Sharma, the legendary author, was hired to make stubborn young royals into thoughtful adults in six months.
The Jataka tales are older. They come from Buddhist tradition and tell the previous lives of the Buddha, sometimes as a monkey, a deer, a king, a merchant. They weren't written for princes. They were told to anyone who would sit and listen, over many centuries.
What they're for
The Panchatantra teaches strategy and judgment. Whom to trust, when to flee, how to read a situation faster than the person trying to fool you. At its core, it's a survival manual.
The Jataka teaches patience and kindness. The path is long. The good heart ripens slowly. Most Jataka stories end with someone making a small, hard choice that turns out, much later, to have mattered.
Both are useful. The order you read them in matters less than knowing what you're handing your child.
Which to start with, by age
Ages 3 to 5: Jataka. The gentler stories with kinder endings sit well with very young children. The animals are mostly friendly. The dangers are mostly soft.
Ages 5 to 8: Panchatantra. By now your child has the appetite for cleverness, a little danger, and characters who occasionally deserve what they get.
Ages 8 and up: Both, side by side. This is when the contrast between the two becomes the lesson.
The blue jackal versus the monkey king
Compare one tale from each. The Blue Jackal (Panchatantra) is a story about being found out. A clever creature pretends to be a king and is exposed when his nature gives him away. The Mahakapi Jataka (the Great Monkey King) is a story about sacrifice. A monkey makes himself the bridge that saves his troop.
Same kind of animal, different worldview. The Panchatantra warns. The Jataka inspires. Children need both.