Apple seldom goes into great detail in its most public description of iOS (and OS X) updates. In this case of iOS 8.4.1, what seems like a minor update is packed with critical security fixes, and all users should update immediately.
Here's the link to Apple's product security mailing list posting that details the fixes. A brief scan confirms that this update should be applied right away.
Apple likely doesn't go into a great amount of detail in its most visible update notes described in iOS Settings > Software Update because it would be tedious and overly technical. Plus, experience has shown that Apple customers are fairly good about updating to the latest iOS version soon after it's released.
Here's just one of many fixes in this 8.4.1 release.
libxml2 Available for: iPhone 4s and later, iPod touch (5th generation) and later, iPad 2 and later Impact: Parsing a maliciously crafted XML document may lead to disclosure of user information Description: A memory corruption issue existed in parsing of XML files. This issue was addressed through improved memory handling. CVE-ID CVE-2015-3807 : Michal Zalewski
Even with prompt update practices, it's not a bad idea to keep track of Apple's Security updates page and be aware of a spouse or child in the family who may have overlooked or ignored the latest updates. Each link in that table, a reformat of the mailing list data, goes into considerable detail about the nature of the security issues, vital information on how exploits could have occurred and what Apple did to fix the problem going forward.
The bottom line? Never ignore even what looks like a minor update that appears on the surface to only address, in this case, some Apple Music glitches.