Both are BDD (Behavior Driven Development) capable unit test frameworks for Scala written in Scala. And Specs is built upon may also involve the ScalaTest frame
The main differences are (mostly from a specs point of view :-) ):
ScalaTest provides more "testing styles" than specs (you can visit each bullet point on the quick start page to get a detailed view on each style)
ScalaTest and specs have a different set of matchers. You can compare them here for ScalaTest and here for specs. On that side of things, specs has a lot of small features that you may like when writing your specification: xml matchers, matchers composition (an easy way to reuse matchers by transforming them), precise failures, detailed differences for long strings, ...
Mockito has been given a nice BDD support in specs: Mockito
specs has DataTables which allow to group a lot of small example in a sort of table (if you can stand operators being used as the table delimiters)
In specs, you can define examples which are nested as libidum and automatically cleaned-up at every level
This is certainly a very partial and biased comparison and many other differences exist (and the libraries are still evolving, ...).
At the end of the day I think that it really depends on your testing/specifying style. If it's simple (simple specification structure, setups, expectations, ...) then both libraries will appear very similar. Otherwise, both have their take on how things should be done. As a last example of this you can have a look at tagging: in ScalaTest and in specs.
I hope this helps.