These instructions work for iOS 8 (tested on both an iPhone and an iPad) using Ubuntu 14.04 (Trusty) and Ubuntu 15.04 (Vivid). The instructions will still likely work in other versions of iOS and Ubuntu, but your mileage may vary.
What's the problem?
If you take pictures with your iPhone or iPad and then plug your phone into your Mac, Photos (previously iPhoto) will import the pictures you took on your iOS device. It will not, however, easily allow you to copy random other photos that happen to be on your iOS device.
We recently had a user come in with an iPhone that had some pictures transferred from another computer with these pictures no longer living anywhere else, and so the user wanted the pictures backed up. Photos and iPhoto do not recognize and import random other photos that weren't taken with the camera.
There are plenty of paid-for utilities you can download that will allow you to copy photos and music off an iPhone or iPad, but if you'd rather save the money for other purchases, here's a little workaround that's cost-free.
Get Ubuntu Linux
Really, you can use any Linux distribution, since libimobiledevice is available for any Linux distribution, but we'll use Ubuntu for this example (and, handily, Ubuntu comes with libimobiledevice installed).
If you're a total Linux novice, you can read up quickly on how to get Ubuntu and how to make a bootable USB of Ubuntu. The instructions are a few years out of date, but the same basic procedure applies, and UNetbootin works on Windows and Mac OS X.
Using Ubuntu to rescue your photos (and music?) off your iPad or iPhone
If you're using Windows, make sure your BIOS or UEFI is set to boot to USB. If you're using Mac OS X, hold down the Option key at boot-up and then choose to boot from the USB stick (it may appear as just the word EFI).
You should get a boot menu similar to the following:
*Try Ubuntu without installing
OEM install (for manufacturers)
Check disc for defects
You want to select *Try Ubuntu without installing
. This will load Ubuntu into your computer's RAM and not affect your hard drive or SSD.
Once the live session is booted, unlock your iPad or iPhone, and plug it into the computer that has live Ubuntu booted on it. Ubuntu will throw you an error saying it can't connect to the iOS device, but that's because you have to say on the iOS device that the computer is trusted. Once you say the computer is trusted, you should be able to select the device from the file manager (you do not need to physically unplug and then re-plug in your device).
To find non-camera photos from a previous sync, go to nameofiOSdevice > PhotoData > Sync > 100SYNCD.
For previously synced music, go to nameofiOSdevice > iTunes_Control > Music. You'll see a bunch of folders labeled F##. Just copy all of them or the entire Music folder that contains them.
For Windows, Ubuntu can write directly to the NTFS drive if you want to copy the photos or music there. For Mac, you'll need a USB stick or something else to copy the photos or music to.