Both could be used and we would understand, but the particle "out" here gives a ring of "very" (I think it´s perhaps more noticeable in spoken English).
However using particles such as out (as in: worn out) up (as in: eat up) etc
adds some spice to your repertoire and the listener will know that you are in a slight higher level.
In the above case it could be interchangeable, tough, without loss of sense.
Yet, it´s not always like this.
For example in the sentence "worn-out shoes with flapping soles" it´s more emphatic using OUT than not using it.
Same goes for "used a worn-out T-shirt for dusting the furniture." would convey better the idea of used to the point of not being useful (for some purpose - here the one of normally wearing it.).
depleted in strength, energy, or freshness <she was feeling very worn after a long day at work>
(see that in this sense, even the dictionarist couldn´t help using the "very" for emphasis?) http://www.merriam-webster.com/thesaurus/worn-out