Well, I am no computer expert, but I think I may have the answer to your question:
>Autosave is a function in many computer applications or programs which saves an opened document automatically, helping to reduce the risk or impact of data loss in case of a crash or freeze. Autosaving is typically done either in predetermined intervals or before a complex editing task is begun.
>It has traditionally been seen as a feature to protect documents in an application or system failure (crash), and autosave backups are often purged whenever the user finishes his or her work. An alternative paradigm is to have all changes saved continuously (as with pen and paper) and all versions of a document available for review. This would remove the need for saving documents entirely. There are challenges to implementation at the file, application and operating system levels.
>Autosave is common in video games. Many video games have an autosave feature that saves progress during a session. For example, in an adventure game, it may autosave after completing a level or mission, or in fighting games, it might save after winning a match. Some games use autosave as the only method of saving data, and the player must complete a set amount of the game before saving takes place.
--Wikipedia on Autosave
Not sure on how to explain this, or if this is a related example, but it can be seen as a comparison to the "Safely Remove Hardware" that is seen commonly on the computers. I've read somewhere (Most probably Quora.com, but I don't have the exact link right now),, where this guy, he explained why one should why "Safely Remove Hardware" is important. Basically,
>If a file is being transferred to/fro a hard-disk to the computer, then it is done so in bits and pieces, as opposed to a single, whole file (same applies to most downloads. Even the websites on the internet load in packages). This saves a lot of time, and makes it easier to recover any lost files. So, sometimes, even after a file has been transferred, some bits of information will continue to be passed around between the 2 devices, an amount so negligible that it doesn't come under the 100% Completed category. Basically, if you unplug the device, this tiny information might be lost. Clicking on "Safely Remove Hardware" basically flushes these bits completely to the source, and makes sure there is no transfer between the devices that can be lost.
Not sure if you read all of that, or to be honest, how it is related, but it does seem to give a peek at the way the digital world works. I may be erred here, but I'm fairly certain that the 3DS and most other devices do the same.
Think of it this way: The "Save and Quit" is the "Safely Remove Hardware", that makes sure all your progress is overwritten to the save file. What happened in your case, about forgetting to push the button, is a common feature: Autosave or Quicksave. Now, both of these are subtly different but for the sake of this answer, and the fact that I am no virtual whiz, let's refer to this gem (from Skyrim's forums, but applicable to most of the games):
>Autosaves happen automatically according to your settings (On waiting, resting, fast traveling, changing location, or having the character menu open for a number of minutes). 3 autosaves are kept. Quicksaves are made on demand, and only 1 backup copy is kept. Regular saves are made on demand, and remain until you delete or overwrite them. Considering how many quest-breaking bugs there still are, you will likely regret not making a regular save every few hours.
So my guess is that the game uses the "Save and Quit" as a proper way to gauge if the player is done playing, or if he/she is merely taking a break. The 3DS then will shut down the game completely, and will cut off all transfer till the next time you open it. This helps it maintain speed, and efficiency, and also ensures that your data is properly saved. No more days of the GBA where your internal battery will run dry, so there's that icing on the cake.
Disclaimer: Since I did not get a perfect, exact answer, I've come up with why I think it functions that way, using general information, and some external sources. If at all you find something that is directly wrong, then please let me know in the comments, and I'll edit it in. Or post a separate answer. Just show some proof of the same, so it is easier to maintain the level of integrity to the answer.