It largely has to do with the geography of the Caucasus. To understand it, we need to understand why other regions did become Islamic and Arabic speaking.

When the Arabs originally unified into a coherent state under Mohammed in the early 7th century, they began to push out into the Middle East. If you look at a topographic map of that area, one of the things you might notice is that the area that is today the countries of Israel, Lebanon Syria, Jordan, and Iraq, are all relatively low lands which are bounded in the north by the Taurus Mountains which are located in the southeast of modern Turkey, and in the east by the Zagros mountains which today forms the border between Iraq and Iran. This area had been divided between the Roman (Byzantine) and Persian Empires, and when the Arabs defeated them in Syria and Iraq respectively, both states attempted to withdraw their forces behind their mountain ranges (Taurus for the Romans, Zagros for the Persians), which was a strategy that failed for the Persians and produced mixed success for the Romans.

Once the Arab/Islamic empire had conquered this area, they attempted to control it by settling Muslim (at this point, almost exclusively Arabic) populations in new cities that they themselves founded, examples being Baghdad in Iraq and Cairo in Egypt. These new centres thrived because not only were they the military and political centres of the new empire, but they were also on the sort of land that the Arabs were used to living on - they had no trouble attracting soldiers for their armies who knew that service meant a slice of the pie. The Muslim population in this area was necessarily pretty thin - it was a huge area and in the 7th and 8th centuries there just were not that many Muslims, but the effect of this was that if you were an elite person living in that area at that time, and you wanted access to political power, you had to go to those centres. Over time, this Arabicised & Islamicised these countries - a process that took centuries. The fact that these lands stayed safe and prosperous as the heartland of the Arabic Empire between their conquest in the 7th century and the Roman resurgence in the 10th century meant that when the Romans (and later Crusaders) did return to contest the lands south of the Taurus, the area was already strongly majority Islamic.

However the situation in the Caucasus was quite different. Prior to the Islamic conquests, the Caucasus had also been divided between the Romans and the Persians. However the actual control that either side had over the area was pretty limited - the land was simply too harsh for these Empires to exercise direct control over. It is full of mountains and valleys and there is no wide open area with the resources to sustain a large state. The Romans and Persians instead controlled it via patron/client relationships with local big men, a fact that was a common cause for conflict between the two Empires as local chiefs and petty kings tried to use the Empires to gain a local advantage over other chiefs. When the Arabs arrived, they found the same problem that the Romans and Persian had - that the land was simply too harsh to march large armies through, so they too elected to make deals with local chiefs who traded their loyalty for autonomy.

As I said before, the Romans had successfully pulled back across the Taurus mountains and into the harsh land of the central Anatolian Plateau (Turkey today). They could not defeat the Arabic armies in battle, but they could fight a guerilla war. The Arab Empire responded by siezing control of the mountain passes throughout Anatolia - with control over those they could raid Roman lands at their leisure, and make sure that their heartlands in Syria and Iraq were safe. However Arab settlement in this area was quite light, both because it was close to the front and so vulnerable to Roman raids, and also because the land was less attractive to the Arabs than lands in Syria, Iraq, or Egypt were. The same was even more true in the Caucasus - the land was just not attractive, particularly considering that the Arabs had a lot of good land available to them further south, and the Romans had kept a military presence there and had diplomatic relationships in the area. The Arabs were therefore content to control the area militarily to protect their more imporant lands, and largely did not attempt to settle it, so there was no significant pressure to Arabicise the area.

When the Turks arrived in Western Asia in large numbers the 10th century, the story was similar. They were pastoral nomads, who were interested in plains which would support their herds. The Caucasus (with the exception of what is now Azerbijan) was bad land for this, so they kept heading west. When they passed through the mountains, defeated the Romans, and made it onto the Anatolia plateau in the 11th century, they found the kind of land they were looking for, and so there they settled, and in a process that was in some ways similar to the Arabicisation of Western Asia, they gradually Turkified the surrounding area. This, of course, would become the heartland of the Ottoman Empire.

I know much less about the history of this region in the Ottoman period, so maybe someone else can speak to that.