Please note that I am not a historian, but simply a person that loves history and especially this area. I may be incorrect and am more than open to correction from a historian.

To understand the answer to this question we must first understand the background. The history of the time period plays a very important role in this question.

Upon the defeat of Austria in the 1866 Austro-Prussian war there was really little chance of Austria ever being integrated into a unified Germany. They were removed from the German Confederation and lost influence over the area. However relations with Germany and Austria from that point started to go up, with plans to build a railway from Berlin all the way to Istanbul which many of the Central powers had interest in. After Germany's defeat of France in 1871 Germany needed allies. France obviously wanted very little to do with Germany after such a defeat. With the defeat of France came a very large upset to the balance of the European continent. A formerly decentralized Germany had surpassed France, the dominant power on the continent, in Military and Industrial power. Instead of a weak Holy Roman Empire Europe now faced a unified Germany, the most populated country on the continent, with a massive economy and a military to match it. If France alone could not defeat Germany, it was not very likely that Germany could be defeated in a 1 on 1 conflict with any other European power as France had a very modern and large military at the time. The United Kingdom grew increasingly hostile as Germany started to build a navy to rival the UK's. The countries had been great trading partners but Britain's survival depended on her navy and feared that Germany could eventually blockade the UK forcing starvation upon the British people. Looking for partners to contain Germany and eventually regain Alsace-Lorraine, France attempted to form an alliance between the United Kingdom and Russia to give Germany a 2 front war, should they attempt to attack France again.

Now back to Austria. As Germany looked for fellow strong allies they were very few options. Japan was strong, but too far away from Europe to have much relevance. The USA was also very distant and isolationist at the time. The Spanish were a good backdoor against France, but elected to stay neutral during the war. Sweden had largely fallen behind the other Great Powers from their former glory in the early 18th and late 17th centuries. This left very few options, with the main ones being Austria-Hungary, Italy and the Ottoman Empire. Austria's mutual borders made it an easy choice, and securing the Ottomans as an ally before the Entente did was very important as it would increase fighting the Balkans with another large front.

So we see that Austria-Hungary and Germany had less of a relationship by choice, but a relationship by force. By leaving Germany no other allies in the region in an attempt to contain them, the Entente really forced increased relations between Germany and Austria.

Now why wasn't Austria annexed by Germany prior to this? Well let's look a bit further back. Austria was defeated by Prussia in 1866. There was little chance of Prussia taking vast amounts of land as it was not a war over Conquest, but instead a war of Hegemony in Germany. In 1871 as Germany defeated France, the other Great Powers released that Germany was now a major player on the World Stage and that containment of what they saw as a growing warmonger was an important policy for main Great Powers surrounding Germany. They had no real chance to annex Austria without the potential of foreign intervention, which could lead to a war with the UK, which would surely be a costly and potentially disastrous event.

Now another thing to consider is fears of Habsburg influence within the newly unified Germany. The Habsburgs were a very powerful family that had dominated the region for a long time and held many lands outside of the Holy Roman Empire and German Federation. By annexing Austria there is then an internal Habsburg influence with a powerful Habsburg Hungary outside of Germany to be used to enforce Habsburg demands.

That's before you even look at the cultural differences in Austrians and what we now consider Germans.

Summarily Austria was a German ally for collective security among Great Powers. Austria was not annexed as there is no way that the European powers would have allowed it without intervention much like that against Napoleon. And finally an internal Habsburg influence with Habsburg domains outside of Germany could possibly threaten Prussian dominance within the new state and threaten the centralization of such a nation.

I would argue that they had a common history, architecture, and they certainly weren't of the same culture. There is a "German" culture but Austria and Austria-Hungry was more defined by Habsburg rule than German identity.

Another problem is that up until it fell apart in the Great War, it was the Habsburg lands making up of a patchwork of different nationalities, ethnicities, and religions. Austria, being the main territory of the Habsburgs, was more of a homeland to the Habsburgs than fully German. So it doesnt make sense for Austria to be engulfed by Germany when they're at the head of their own Empire. I cannot say of the post Great War Austria though.

Since post-1866 is covered by realffagore, I'll try to go a bit before that as a lot of it has to do with Austro-Prussian relations, and can hopefully explain why Austria wasn't part of Germany in the first place.

Austria had very little desire for unification largely due to the influence of its Foreign Minister (and later Chancellor) Metternich, who was an extremely reactionary and conservative individual who wished to maintain the state of Europe based on the Treaty of Vienna (1815).

Prussia, on the other hand, had ambitions following the 1848-49 revolutions, and this led to the end of "peaceful dualism" where both Austria and Prussia respected each others' influence in Germany and began a period of obstructionalism between the two states. Bismarck is quoted as having said of this "Germany is clearly too small for us both", and so with the question of having either a Großdeutschland (Germany with Austria) or a Kleindeutschland (without Austria), the latter became preferable, especially with Bismarck as one of the leading forces behind Prussia's actions.

Plus there are other things to consider, like the state of Prussia/Germany compared to the Austrian/Austro-Hungarian empire. Prussia was a wealthy nation with a large amount of raw materials and heavy industry in areas like the Ruhr, making it an industrial power which eventually surpassed Britain. Austria, however, was mainly agricultural and the empire was virtually bankrupt by the Austro-Prussian war. So overall including it in Germany would not only have had the problems of Hungarian/Czech/Croat etc resistance but also it would have had a huge negative effect on the German economy.

Furthermore, Austria had interests in the Balkans, whilst the Prussians/Germans at the time of unification cared little, with Bismarck saying that they weren't "worth the bones of a single Pomeranian grenadier", and before this he referred to the Austrian neutrality in the Crimean war as an attempt "to procure a few stinking Wallachians", clearly showing a conflict of interests between the unifying forces in Germany and the Austrians.

Source: "The Unification of Germany 1815-1919" Third Ed., Farmer and Stiles