Many people have a love-hate relationship with iTunes—and for some it's more hate than love—but it turns out It’s entirely possible to keep your device backed up and to manage files effectively without using iTunes at all. I found three Mac apps that let you manage many aspects of your iPhone: iExplorer from Macroplant, iMazing (which used to be Disk Aid), and Ecamm’s PhoneView.
There are many reasons why iTunes can be more of a hindrance than a help. Perhaps you don’t spend much time at your computer so syncing your iOS devices to iTunes isn’t a useful way to keep your device backed up, or maybe you don’t use iTunes much so having to spend time learning everything again whenever you do remember to sync your device is kind of a pain. Or you are only interested in copying over photos, for example. (Note: I’ll be using iPhone and iPad interchangeably since both are shorter than “iOS device” but the rules are the same for both.)
iMazing
IMazing (US$29.99, code MGG saves 20%) has a “cloning” feature and a backup browser, so if you just want an exact copy of your iPhone “in case,” you’re all set, and if you just need that Angry Birds game settings file or an app that’s no longer in the App Store, you can copy that out for safekeeping too.
Within iMazing’s interface you can transfer music both to and from your iPhone easily, even copying the iPhone music to a specific folder instead of into iTunes directly. If you like iTunes as a music player, you can even allow iMazing to import your iPad playlists, it will recognize the tracks that do and don’t exist in iTunes, and copy over the ones that aren’t there.
Photos can be copied easily, including shared Photo Streams and videos or panoramas or the entire Camera Roll. Messages (including MMS), call logs, and Visual Voicemail can be scooped up off your iPad and copied over to your Mac, even allowing for export in a few different formats depending what you need.
iMazing's user interface window. Is that really Jony's iPhone?
iExplorer
Macroplant’s iExplorer (limited demo, full version $35) offers a “disk mounting” feature which treats your iPhone like a hard drive connected to your computer, visible in the Finder and treated just like any other disk, allowing for single/specific file transfers.
IExplorer offers similar app and music management to iMazing, with the ability to build playlists and copy only the songs that aren’t on the Mac, or if you just need to transfer all the music from your iPhone to your computer, it can handle that too.
You can copy photos off your iPad and into whatever folder you like, or straight into iPhoto if you prefer. Messages/Voicemail can be exported in different formats, along with call logs and pictures/videos sent via MMS.
iExplorer's main window. Shouldn't take long to back that thing up...
PhoneView
PhoneView (limited 7-day demo, $29.95) allows for viewing and accessing virtually all the media on your iPhone, offering archives you can browse even when your iPhone isn’t connected. Music and any other media file can be copied off to your Mac easily, whether it’s in a batch or individually.
Photo management looks much the same as other file browsing in PhoneView, allowing copying of images to and from your iPhone easily. Messages/call logs are accessible and can be set to back up automatically each time you connect your device. Other messaging apps and voicemail can be archived as well, with messages exportable to PDF if you want them outside the archive in PhoneView.
PhoneView, showing individual files stored in an app.
Summary
All three of these apps are easy to use, so whether you need an easy way to manage music or some serious file management, you should be well-served by any of the three listed. If you want a straight up direct copy, iMazing is your best bet.
An advantage to these apps is that they also don't require that the device be tied to any particular iTunes library so if you need to help someone rescue their device, you can hook it to any of these apps and get the info you need without worrying about authorizing computers or erasing libraries or transferring purchases. In any case, having one of these handy will improve the syncing experience by quite a bit.