I'm not sure what the benefit of only having one apprentice is, maybe a time and attention type of thing?
- For the Sith, it was self-defense / keeping your apprentices from killing each other.
- For the Jedi, it was better for training in a 'Way', and keeping close watch to keep them from sliding to the dark side as their skills develop.
It's a different motivation for Jedi and Sith.
For the Sith, it was
to keep there from being some much infighting between the apprentices, as you saw before the Rule of Two was instituted.
to avoid the possibility of them ganging up on the Master.
Bane learned from Revan's holocron that, in times past, students of the dark side who were impatient and hungry for power would make alliances with his Master's other apprentices in order to defeat him and take the mantle of Master.
The Jedi, on the other hand, didn't really limit it until the student had reached a certain point. (Remember the Younglings that Vader slaughtered? All in class together.)
But once they reached a certain point, it was important to maintain a very close relationship between the Master and the Apprentice, as the temptation to slide to the Dark Side could get very strong as one gained more power.
The training was more than just in the Force and tactics, at this point, it was an attempt to train an entire way of life; that worked better by having the student constantly with the teacher, and that they could focus entirely on each other when it came to learning. (Students would rarely learn the same things, at the same pace.)
Some extra History (courtesy of Wookieepedia)
At some point after the Great Sith War, the High Council was first convened and set in place the bylaws of the Order. During this period, the term Padawan was introduced and Masters were restricted to taking only one at a time. Over time, as the Order flourished, the High Council decided to enforce even stricter standards such as a cut off age for Initiates to be selected for apprenticeship. At the Jedi Temple on Coruscant, which housed the largest of the academies of the era, an annual Apprentice Tournament was created to not only test the students at the academy, but also to aid in the selection of apprentices by prospective masters. While it set the rules of the Order, the Masters on the High Council typically avoided direct involvement in this selection, preferring instead to allow the Force to bring a Master-Padawan pair together
For the Jedi, one might pay attention to the word "knight". Traditionally, European knights had a single squire who served as an apprentice (although having more than one wasn't uncommon, if the knight could afford it). The Spartans had a similar system, where as part of the agoge, a Spartan boy would be paired up with a young Spartan man, which was, the pederasty aside, served as an instructional tool for the future Spartan soldier to learn the tools of the trade and become indoctrinated into Spartan culture.
Having only one at a time allows focus on instruction that isn't really possible with multiple apprentices.