It is always a little risky to say that two sentences mean exactly the same thing. These two are pretty close. But the thing to keep in mind is that supines in Latin are rare in general; and they are hardly ever used with anything other than a verb of motion, usually ire or venire. (For the standard discussion, see Allen & Greenough 509, Bradley's Arnold 402.) Scribere might perhaps be considered a verb of motion, if you include the ideas of writing the letter and also sending it. But one would have to search the corpus of Latin literature to see if it is actually used this way.
All that to say, the two sentences might mean the exact same thing, but the one with the ut-clause would sound "normal" to a Latin-speaker, while the one with the supine would probably sound "weird."
Dic mihi, Damoeta, 'cuium pecus' anne Latinum?