I am not aware of any advantage, but my family and friends are using either Threema, Signal or Telegram. So I use all three of them.

Why Threema?

Currently limited to groups chats of 256 participants and group calls of 16 participants

For privacy, freedom, and control, iOS is out. As with Telegram, I advise staying away from the Google variant and highly recommend the Threema Libre implementation for Android. Licenses are not compatible across variants, so stick with the Threema Shop!

• Open source full E2EE (end-to-end encryption) platform with regular security audits
• Mobile clients can be linked to Threema Desktop/Web
• Source code available for Android, iOS, Desktop, and Web
• Threema Libre implementation for Android
  • Liberated from Google Services, uses Threema Push for instant delivery
  • Reproducible builds
  • Available via F-Droid repository
• Simplified (normal human-understandable) Changelog
• Multiple ways to purchase through Threema Shop
  • With or without Email Address
  • Cash, Bitcoin, Mastercard, Visa, or bank transfer - Can leave every non-payment field blank!
  • 4.99 CHF, EUR, or USD per license
  • Pay anonymously with cash (rounded nearest 10, buy in multiples of 2)

The largest criticism levied against Threema:

• 2023010? - Accusation - Three Lessons from Threema - Analysis of a Secure Messenger
• 20230109 - Rebuttal - Statement on ETH Findings
• 20230110 - Coverage - Messenger billed as better than Signal is riddled with vulnerabilities

"While some of the findings presented in the paper may be interesting from a theoretical standpoint, none of them ever had any considerable real-world impact," the post stated. "Most assume extensive and unrealistic prerequisites that would have far greater consequences than the respective finding itself."

Its business model and not requiring a phone number

Titanium Backup hasn't been updated in five years, and I think that update was just to meet requirements to stay on the store. Their last changelog entry is adding the menu icon after Android ditched the physical menu button. There are a bunch of settings that are broken or do nothing due to changes to Android over the decades (TB has been around for so long that it supports Android 1.5).

I've been using Swift Backup as a replacement these past few years. It's closed source but was recommended to me, and I haven't run into any problems yet. Is Neo better in some way, aside from being FOSS?

Titanium has been abandoned for years. Swift backup is what I switched to, or DataBackup on GitHub if you want a free option.