Full disclosure: I can't name the orchestra when a tango comes on. I can, however, hear the differences between versions of a song - particularly one that I like a great deal. If I had a better music vocabulary, I think I might have an easier time remembering the names because I could attribute certain characteristics between version and orchestras. As it is, I have my own little internal dialogue that goes something like, "oh this is the version that's slower in the beginning then speeds up." Or, "this is the one that stays slow and melodic, but ends really abruptly."
While I was trying to explain the importance of listening to several versions of songs to get an idea of the range of the music, I thought of an example that seemed to almost make my point for me. So to the tanguero who asked, this is why I think the orchestra is important. . .
Here is Eric Clapton (Cream) song Layla (1983) with Clapton, Jeff Beck, and Jimmy Page:
And, here is Clapton and Mark Knopfler doing the same song but much more slowly - a very different energy and feel to it. To really express the music, you'd have to dance them differently.
They're both great pieces of music but evoke, for me anyway, very different feelings when listening to it.
Now take El Choclo, one of the most recognizable pieces of tango music. First check out todotango's page for the background of the song: http://www.todotango.com/english/las_obras/Letra.aspx?idletra=24
Now, here is Tango el Choclo, Orquesta de Francisco Canaro:
And here is Piazzolla's El Choclo:
And if you're really interested: Nat King Cole's Kiss of Fire for comparison: