The plight of the Austrians in the grand global history of civilisation is a vastly different story to that of the Holy Roman Empire, though the two did intertwine on several occasions.
The Holy Roman Empire (roughly the location of modern Germany) consisted of a billion and one (exaggeration for humor) small princedoms, kingdoms, city states etc which, every now and then, had a vote to decide which dude was going to be the next Emperor. The Emperor would have the luxury of fund raising for war and installing their marital wannabes in the right places to generate more real estate upon the death of the correct princes/kings etc.
The area called Austria was one of the subjects of the Holy Roman Empire. However, over time, the rulers there got increasingly annoyed at the fact that they did not permanently rule all of Germany and often felt a tad upset whenever the rest of Germany snubbed them in favor of some dimwit from Lesser Ballsteinwick.
This lineage of disgruntled Austrians was pretty unceasing and the name Hapsburg soon overtook the name Austria when people referred to this land. Like the British and their relations with Europe, the Hapsburgs had started building a huge empire well away from the borders of Germany and now felt that their membership of the Holy Roman Empire was more a token of goodwill than an order of faith.
It wasn't long before people now referred to this area as the Austro-Hungarian Hapsburg Empire. They picked up lands from the Turks in Eastern Europe, including the infamous Transylvania, lands from the south where Venice continued to recede, Mediterranean Islands, Chunks of the Netherlands and, at their peak, took the entirety of Spain (during which time the famous Inquisitions took place).
The capital of this Empire, Vienna, became a cultural icon on par with Paris or Rome and the very essence of civilisation was considered to be, at heart, Austrian, as is the case with all civilisations at their peak. As the old saying goes, however, the bigger they come the harder they fall, and, when a Serbian Rebel shot the Arch Duke, the First World War was initiated and the Austrian Empire vanished from the globe.
Right from about 800AD to present Austria has had military ties to the Holy Roman Empire (or Germany if you prefer) and in the event of external threat this area always has and always will act as one, but, beyond that, to say that the Holy Roman Empire is the same as Austria would be like saying Canada is the same as the United States of America, or that Venice is the same as Italy etc.
Hope that's helped clear that one up a bit!