Unfortunately for this discussion, the OP has made assumptions that are mutually exclusive. He assumes that the two races have the same technological base, then asks what we'd do now? Sorry, it doesn't work that way. As has already been pointed out, we'd almost certainly make radio contact with them in the 20s or 30s. From that point on, everything changes. Von Braun was amazed when the allies asked where he'd gotten his ideas from. His answer? "From your Dr. Robert Goddard!" I would also bring to your attention the Disney 'Man Into Space' series from the 50s. If the cold war space race hadn't happened we'd be on Mars now (using atomic powered ion drive ships), with space stations and probaly a Moon base as well. So we'd have a whole different set of space technologies than we we do now when that alternate reality got to the 21st century.
I'd also like to bring your attention to a book writen by H. Beam Piper and someone else titled First Cycle, which considers exactly this scenario, although the two planets orbit each other. (Yes, they wipe each other out.)
As for the first contact story that was mentioned, I believe the title is, in fact, First Contact. By Murray Leinster? It's easy enough to look up, my potential confusion is because there were a number of first contact stories published around the same time. But, for obvious reasons, that's the most well known.