From the ESV Wikipedia link (emphasis added):
The result is a translation that is more literal than the popular New International Version, but more idiomatic than the New American Standard Bible.
That's probably the main difference. So ESV is going to be more literal, less figurative and free from exaggeration/embellishment in it's translations and is going to "use, contain or denote expressions that are natural to the native speaker" (idiomatic as defined by dictionary.com).
(My opinion) From what I've read from each of them, the NASB seems to be smoother and more easily understood in modern English. However, with the ESV, it's nice sometimes to get a more clear look at the original meanings/translation.
The most updated version of the NASB was 1995* and the most recent version of ESV is a 2011 edition*.
"[...] on occasion the ESV translates 'person' or 'one' where previous translations used 'man' [however] it keeps gender specific language [as-is] in the original [text]."*
*from Wikipedia links provided.
Good question, I learned a bit through this short bit of research :)