You may be interested to know that both, Slovaks and Slovenians, refer to their respective countries as Slovensko. Just thought I'd confuse you a little more. :)
Seriously, before the Hungarians came to Europe (9th Century, or so), Slovakia and Slovenia were not separated, but occupied continuous land. They probably did not think of themselves as separate ethnicities.
To this day, Slovaks and Slovenians are two friendly nations. Their languages are quite similar.
By the way, if you want to take a plane to Slovakia, your best bet is to actually fly to Vienna in Austria, then either take a bus to Bratislava (the capital of Slovakia), or rent a car and drive. The airport of Vienna is only about 25 miles from Bratislava.