Turning into a Great Ape is an important ability for a Saiyan in Dragon Ball Z, and something that several Saiyans have done to turn the tide when losing a fight, so why is it that Trunks and Goten don't have this power? The key to unlocking this ability is of course the Saiyan tail, which Trunks and Goten don't have. But why?
All Saiyans are born with tails, which allow them to transform into a giant monkey called a Great Ape. In this form, the Saiyan is unable to control himself -- with the sole exception of Vegeta -- and is usually depicted as an unstoppable force. Great Apes have been present in the anime since the first series, Dragon Ball, where it first happened to Goku. In this form, Goku accidentally killed Grandpa Gohan. Later, the form was used by Gohan while training with Piccolo. Vegeta used this ability to gain an edge over Goku, and nearly defeated the heroes thanks to his massive boost in power. Everything changed when Gohan too accessed the Great Ape transformation.
The Great Ape form was an important part of the Saiyan Saga, but it faded away for the remainder of Dragon Ball Z. This made sense at first, since Goku, Gohan, and Vegeta all lost their tails, but then the Buu Saga came, and Kid Trunks and Goten were introduced. Since neither character had tails, the anime never explored their own experiences with the Great Ape form. They were half-Saiyans just like Gohan, so it stood to stand to reason they would have this trait too. Fans have often speculated on how this could be. Some have suggested that their parents had their tails cut off off-screen to avoid conflict. A popular myth alleged that series creator Akira Toriyama simply forgot to include the tails when Trunks and Goten made their Dragon Ball Z debuts.
However, both of these theories are wrong. Akira Toriyama explained in an interview (via Kazenshuu) that the tails were dropped from the anime because Toriyama never understood how a Saiyan could get dressed while having a tail. Figuring out how this would work was a "real pain", so Toriyama stopped using them.
It's worth noting that Dragon Ball Z no longer needed Great Apes, so not including the tails was no major loss for the series. Great Apes were powerful opponents in the Saiyan Saga, but the main characters grew in strength as the series went on, and it reached a point where it felt as if Dragon Ball Z had "outgrown" the Great Apes. The Great Apes could be another example of a concept in Dragon Ball Z that was once a great threat, but has since been left behind.