From one of their comments at GitHub:
The core difference is that the design of Truth includes two specific
areas of extensibility - that of a strategy for proposition failure -
such that a "subject" for Integers, or a subject for Strings can be
re-used in the context of completely different results for failure. A
notable example is the distinction between JUnit's use of
AssertionError and it's AssumptionViolationException. Truth lets you
use the same proposition classes for both.
The other area of flexibility is the ability to create new
assertion/proposition types and hook them in without declaring
possibly conflicting static methods to import. This can be for new
types (say, adding protobufs) or for new uses of existing types (say,
Strings that are treated as Uris). This is the assertAbout() feature.
Other than that, Truth is very similar to AssertJ, since it was
inspired by FEST, of which AssertJ is a fork of the 2.0 development
line.
To sum up, Truth is designed to be a bit more extensible and flexible, but AssertJ will be great (possibly the greatest) for assertions on standard types.