I went to a rockabilly live the other day and I saw that the double bassist used transparent plastic strings on his bass.
What's the purpose of that? Is there anything different from the normal steel ones?
It is likely an attempt to get closer to the sound of gut strings that were used on the double bass in times of yore (and now, although it's not so common). The slap style, and the back to basics approach of rockabilly would suggest this.
In addition to the gut string feel cited by Meaningful Username in her/his answer, another advantage of strings without a metal core is that they are less susceptible to changes in pitch resulting from climactic change, especially temperature. At least that is what this manufacturer claims. The Tremors are one of the bands who use their strings, and they look pretty rockabilly to me: