I can't find this information documented anywhere.
What filesystems do iOS devices use?
Did Apple stick with HFS+?
Is it different between devices and iOS versions?
As of iOS 10.3 (March 2017) - APFS/Apple File System is now the file system being used.
Previous to iOS 10.3 HFS+ was being used.
APFS will add improved support for solid-state drives and encryption improvements - things that were not a priority 20 years ago when HFS+ was introduced.
The other answer seems to be correct about the file system being HFSX*, but on being case-insensitive (other answer since edited to correct), I found that not to be the case in the book Mac OS X and iOS Internals. On page 23 it says:
In iOS, being the case sensitive HFSX by default, case is not only preserved, but allows for multiple files to have the same name, albeit with a different case. Naturally, case sensitivity means typos produce a totally different command or file reference, often a wrong one. [emphasis above is mine]
Since that didn't agree with the other answer, I sought confirmation. I loaded a sample iOS project I was working on and changed the name of the database to only differ by case. After running the app on my iPad mini (iOS 6.1) to re-create the database, I went into iTunes and looked at the files under File Sharing for the application. Here's what I saw:
So, it would appear Mac OS X and iOS Internals is correct: case is preserved and allows for multiple files to have the same name.
Interestingly, when I selected both files in iTunes' File Sharing and clicked "Save to..." my desktop, I did not get any warning. iTunes apparently "saved" both files to my desktop, but only the initial-uppercase version survived (presumably overwriting the initial-lowercase version of the file.)
As of iOS 10.3, released on March 27, 2017, iOS now uses the new Apple File System (APFS), said to be "optimized for flash and solid-state drive storage, with a primary focus on encryption."
APFS was announced at WWDC 2016 and there is a WWDC video titled Introducing Apple File System, and an Apple File System Guide for developers. The features section of the guide may be of specific interest.