They can provide some significant (i.e. measurable) performance improvements on low-end devices, where things like allocations (e.g. for-each loop) or auto-boxing can be very detrimental.
For example, LibGDX's StringBuilder.append(int) doesn't lead to any allocations, which makes it especially attractive for displaying integer text on Android (i.e. FPS, player score).
If you are building a 3rd party API, it would be poor practice to use, say, LibGDX's collections instead of the standard collections. But if you are building a game, or a LibGDX-specific extension, or working on internal code that needs to be highly optimized, then it makes much more sense to use LibGDX's collections.
Even the Android docs recommend against getters and setters, which makes the LibGDX utilities desirable:http://developer.android.com/training/articles/perf-tips.html
Aside from the collections, the vector/matrix and math utilities are really excellent. The vecmath is in many areas accelerated with native code. And the MathUtils sin/cos/etc are usually a little faster than java.lang.Math, at the expense of precision.
And then there is a whole slew of other utilities, like reflection (that works correctly with GWT), Pools for re-using temporary objects, fast and lightweight JSON and XML parsing, etc.
TL;DR - the GDX utils are the best thing since sliced bread, so definitely use them in your games.