How similar are the Turkic languages Azeri / Azerbaijani and Uzbek?

This article has the following:

In his turn, calling Azerbaijan as fraternal country, President of Uzbekistan noted that the two countries were close both historically and culturally. “It is difficult to find in the world two peoples tied with each other by their roots. We can talk in our native language and we will always understand each other”.

That's pretty much the main gist of it, that people from both countries have quite an easy time understanding each other. I would suspect that these ministers would have a much easier time understanding each other through all the interaction they've already had with one another and other Turkic countries, giving them an instinct for which words are common to all and which aren't in the same way that Scandinavians more or less know which words to use and which to avoid when talking with one another.

Azeri and Uzbek certainly have a lot of terms common to each other that don't exist in standard Turkish though. Words like:

• az men, uz men, tr ben (I)
• az kömäk, uz kömäk, tr yardım (help) <-- az and uz also know yardım
• az barmaq, uz barmoq, tr parmak (finger)

But on the other hand there are also quite a few words that are common to Turkish and Azeri but don't exist in Uzbek. In any case, the languages are all quite similar and the general rule is that geographic proximity also means linguistic proximity, but it's not a hard and fast rule (see the Salar language for example, located inside China but much closer to standard Turkish than all the other languages that lie in between them).

Also note that Turkish is the only Turkic language that has a dominant standing in the country in which it is located. Azerbaijan is the closest to this but the population is small enough that not many people learn Azeri, and all the other countries that use Turkic languages are still working on establisting their own languages under a lot of Russian influence. Microsoft Office for example has finally been translated into Kazakh; until now people there have had to make do with a Russian language version. Uzbekistan is still getting used to the Latin alphabet and Kazakhstan might change over to the Latin alphabet soon, so the only real stable language in all this is standard Turkish. The situation for these languages is of course much better than it was a few decades ago when these countries were a part of the Soviet Union, and they had no common script, which aside from a few holdouts such as Kazakh and Kyrgyz, has become the Latin alphabet. Taking a look at this image for example one can see that Turkish + Azeri + Uzbek + Turkmen = almost 80% of the total population. Tatar is interesting in that it uses the Cyrillic alphabet because it's the law in Russia, but their Wikipedia is in the Latin alphabet.