There is the famous phrase "Iki devlet, bir milliyet" (or "two states, one nation") that has been used to describe the relations between Turkey and Azerbaijan, but more interesting are the cultural and social similarities and (more commonly) differennces.
To begin with, the language. By and large, the two languages are relatively similar: as similar as, say, Dutch and German, or Norwegian and German - but NOT as similar as, say, Norwegian and Swedish. This has been a cause for both Turks ans Azeris when using their native language in the other's country. Turks come to Azerbaijan speaking Turkish may well be pretty easily understood by Azeris, but as Turkish grammar forms are more various than Azeri (though Azeri has its own various, but differing grammar structures) they will instantly be identified as non-Azeris. One interesting explanation I heard is that Azeri structures bear more similarity to old Ottoman structures, that have since changed much over time in Turkey itself. In other words, when Turks hear Azeris speak, it sounds like they're speaking an archaic form of Turkish. The letters are different in each language, too, and, much more interestingly (and open for entertaining misunderstanding), are the differences in vocabulary.
To give just one example: the Azeri phrase "stop the bus, I want to get off", using the same words in Turkish translates as "hide the bus, I want to fall".
Another reason for the differences is also historical. As Azerbaijan was occupied by Russia for most of the last two centuries, many everyday words have been replaced by the Russian ones (a similar point could be said of Turkish when considering the effect that French has had on their language). Also, there are many leftover Persian or Arabic words in use in Azeri, more than in Turkish, from what I can tell.
In Azerbaijan, and Baku in particular, Russian is still the "imperial" language; like English in today's India, perhaps. Anyone who doesn't have at least a working knowledge of Russian in Baku can be looked down upon, in particular in officialdom or in respectable companies. There are still a large number of Azeris in the country who speak Russian as their first language, and it is almost essential to know at least some Russian to get certain level of intellectual respect from one's peers.
On the other hand, a Russian speaker who has no or limited understanding of Azeri (and one of my former students could be counted as one) struggles to get by with the average Joe on the street. This is Azerbaijan, after all, and not Russia, or even Kazakhstan (where Russian speakers are still a huge segment of the population), and a Russian speaker ignorant of Azeri could be considered elitist or snobbish by the same officialdom and respectable companies. As you can imagine, a fine balance seems to required, but that is where most people exist; seamlessly interchanging between Azeri and Russian mid-sentence throughout conversations.
This has a naturally direct effect on the culture. For Turks who consider Azeris as their linguistic and cultural kinsmen, it comes as a surprise when they find out that they are just one of three competing cultural influences on Azerbaijan: the others being Russia and Iran. For this reason, the cultural differences are subtle but many.
To begin with, contemporary Azerbaijan appears even more secular than Ataturk's Turkey. In Baku, the call to prayer is only heard in the old city, where almost all the historical mosques are. The only other mosques in Baku are relatively new ones built in a few working class neighbourhoods near the centre, or in out-of-the-way suburbs that belong to very devout Muslims. The city centre and much of the city appears practically religion-free. The legacy of Soviet Russia is an obvious explanation for this. The comparison to Turkey, which is also officially secular but has a muezzin-blaring mosque on almost every corner, is stark. The historical reasons for the situation in Turkey are also more complicated: Ataturk never succeeded in fully dampening the devout tendencies of some Muslims in Turkey; he only succeeded in postponing them. The rise of the AK party over the past decade is a testament to the fact that these Islamic sentiments were always just under the surface, waiting for the right opportunity to be returned to the forefront. But that's another story. Perhaps because pre-Soviet Baku was also the most cosmopolitan city in the Russian Empire is another reason why devout Islamism has seen only a fairly modest revival in Azerbaijan overall, and that devout Muslims are a fairly marginalised bunch that few modern Azeris have time for.
But don't let the ostensibly secular appearance of Azerbaijan make you think that this automatically applies to the mindset, though. Unlike Turkey, where the headscarf is prevalent (moreso in recent years), in Azerbaijan it is much less common, for the reason that it has a much stronger link to devout Islam. But the lack of headscarves distracts from other, more socially-conservative tendencies in Azerbaijan. Although there are signs that things are changing (more on that later...), being in an openly sexual relationship before marriage has been considered socially unacceptable in Azerbaijan. Azeri society still has largely traditional views of men and women - to the extent that for many women, the pinnacle of their life's achievement would be to have a good (rich) husband, healthy children, a good house etc etc. And naturally, many of the men expect the same of their women.
To be fair, these views are hardly unknown in the West, either (think of Greece or southern Italy), and there are these days plenty of young women in Baku beating this stereotype with successful careers, and men who have more enlightened views towards the opposite sex. Also, even more encourgingly, the number of young Azeris who have now lived abroad (and have returned to Baku with their Western mindset intact) is increasing all the time, and the streets of Baku are these days filled with contemporary young people. Baku city centre these days looks as much a wealthy Western city as Paris or Copenhagen, albeit with a slightly Middle Eastern flavour.
Then there are more everyday cultural issues such as clothing. Although Turks as shown in the media world dress to kill, and office workers dress to impress, the average Turk in the street seems to dress to...well, dress. In other words, casual clothes, jeans, T-shirts are the norm. Turkey is a young country with a huge youth culture - it seems they are too cool to feel the need to "impress"; much better to wear whatever you feel comfortable in. In this way, young Turks in particular seem like a throwback to eighties America; casual is the new smart.
Azeris, by comparison, certainly dress to impress, every single day, and especially when out with family and others. Azeris are much more conscious, it seems (as Italians are, perhaps) of looking good for the sake of others. Going to a smart cafe in your best clothes is totally normal to an Azeri; they would never consider otherwise. In this way, it may appear to make Azeris more conservative that their Turkish counterparts; maybe, but "conservatism" has many different meanings and expressions, as we can see from the different ways that Turks and Azeris express their culture.
In Baku what was only two years ago unheard of is now commonplace. As I said, Azeris tend to dress to impress; young Baku women these days dress to impress in more and more openly obvious ways. While Turkish women are still happily "sexy" in their T-shirts and jeans, young Azeri woman wear high-heels (though high heels are common for all ages of Baku women), short skirts (which were more linked to prostitutes in Baku in the past, though they still exist) and tonnes of makeup. From what I've seen, Turkish women tend to be much more indifferent to the use of makeup (again, "casual" is king in Istanbul). Whereas Turks may get their clothing and style inspiration from America and the West, Azeris tend to get it from Russia and Iran (Iranian women, even in a headscarf, are obsessed about wearing the right makeup).
One last thing: religion. Some Turks don't realise this, but whereas Turks are Sunni Muslim, Azeris are overwhelmingly Shia. This has an effect on the culture again, but also in subtle ways. The best way to sum it up would be to say that Azeris are more fatalistic as a nation than Turks. Turks have character flaws, too (what nationality doesn't?): a kind of whimsical melancholy that comes from the national character and history; a tendency to see things in black or white; but I don't want to linger on them all.
On the positive side, the long tradition of multiculturalism still exists in today's Azerbaijan to a healthy extent. This is also because nationalism is still quite a new concept to most Azeris. "Azerbaijan" as one empire never existed for long in history as a single entity; most of the time there were rival miniature kingdoms under pressure from Persia or Russia, which were all gone by the early 19th century.
Turkey's sense of self is much more obviously nationalistic, and that has been to the general detriment of multiculturalism. Like Azerbaijan, "Turkey" is also a comparively new concept, created out of the mind of Mustafa Kemal ("Pakistan" was created in a similar fashion after World War Two). Previously it had been the Ottoman Empire, where, although it was a Muslim empire, it was also hugely multicultural as it contained vast numbers of Christians, Jews and others. Many of these "minorities" had a huge cultural impact (and influence) on the Ottoman Empire. The creation of Turkey made all these people "Turks". The elephant in the room was that "Turkey" still contained many "minorities", and although the country was officially secular, in practise, it could be said that Turkish nationalism overlapped with cultural imperialism.
Nowadays, Turks are very protective and defensive of their identity (perhaps Azeris naturally will be too, after another decade of oil-fuelled confidence?), and this has made cultural re-opening a sensitive issue. Some commentators talk of the AK party's policies as a "New Ottomanism" with more engagement with the regional players and neighbours, and a re-flowering of multicultural life inside the country. The former may well be true, but the jury's still out on the latter. As for Azerbaijan, they're still playing their role in the Great Game, now with the added impetus of oil and gas to back them up.
Both countries are going places; just not necessarily to the same destination.