Both phones come loaded to the teeth with apps out of the box, but all apps aren't born equal.
Take the music player for example - the Samsung Galaxy Note 3 packs equalizers, SoundAlive, Smart Sound (equalizes volume between tracks), Adapt Sound (tunes the EQ automatically to your headset) and FLAC support (16 and 24 bit).
The HTC One Max lost the BeatsAudio setting, but HTC didn't replace it with a proper equalizer. Sure the One Max has the better speakers, but most people use headphones for music - in the next chapter we'll see if HTC has kept the quality audio hardware. The player supports FLAC files (16 and 24 bit).
By the way, the One Max has FM Radio, which is still an important feature to some.
The Galaxy Note 3 video player handled every codec and container we threw at it, while making good use of the advanced sound options. It has the Pop Up play option too. The HTC One Max has limited codec support, so you'll have to rely on third party apps.
While we're on the topic of multimedia, we should note that both devices can output video over DLNA and MHL. The Note 3 supports MHL2, while the One Max uses the original MHL standard. The major difference between the two is in the kind of adapter required. Some TVs support MHL without adaptors, so you should check if yours supports MHL or MHL2.
Both phablets have IR blasters too, to control TVs, set top boxes, DVD and Blu-Ray players and more.
The HTC One Max has the edge here as its IR blaster can learn control codes from existing remotes, while the Note 3 cannot. This means you can control practically anything that uses an IR remote, even if it's not in the app's database.
The One Max also comes with an Office document editor, while the Galaxy Note 3 is bundled with just a viewer. This seems like a major omission on such a business-centric device - after all Samsung crowed left and right about how awesome the KNOX enterprise security feature is, but what good it is when you can't edit the Word file you just received?
At least the Samsung Galaxy Note 3 comes with a capable file browser, which the HTC One Max does not have.
The HTC One Max (like other HTCs) comes with a handy app that will transfer your contacts, along with other kinds of data stored on your old phone - Android 2.3+, iPhone, BlackBerry, Windows Phone and even some feature phones.
The two phablets have custom note taking apps - Scribble on the HTC One Max and S Note/Scrapbooker on the Samsung Galaxy Note 3, which can sync with Evernote. The Note 3 does have the S Pen, which makes taking notes significantly easier. HTC has an aluminum capacitive stylus for the One Max, but that's round and easy to lose, especially since you can't sheath it into the device itself.
Samsung also has several apps that might have limited application but seem fun nonetheless. You can line up several Galaxy devices and play music on their speakers simultaneously or connect several Note 3s into one large display.
The S Health app is a great exercise tracker too - it uses the phablet's own sensors but can also connect to external devices like scales for a more complete picture of your health and progress.
Winner: Tie. Both phablets have something the other doesn't and while the Galaxy Note 3 has more apps, some of them are fairly limited. No document editor is a big deal (those are usually pricy), while no file browser is not bad. Plus the remote control app on the HTC One Max is pretty nice.
The Note 3 has an equalizer and better codec support, but no FM radio. S Health is nice, but some may find the data transfer app on the One Max better (who's going to run with a phablet?).