How similar is Azerbaijani to Turkish?
I would say Azerbaijani and Turkish is mutually intelligible. Both languages are the part of same language family - Turkic language family, and both languages are in Oghuz branch (specifically West Oghuz branch). Only Turkmen is considered East Oghuz branch of Turkic language family. But question can be rephrased as "How similar is Azerbaijani Turkic to Istanbul Turkish" - because there are some areas of Turkey like Erzurum or Kars, their Turkish language is practically no different that Azerbaijani, just called differently for political reasons.
So how similar or different is Azerbaijani and Istanbul Turkish? it's not much different, grammar is 99% same, and vocabulary is more than 80% same. The difference comes from 3 points, first Turkey had a language reform - where some scholars tried to replace arabic and persian loanwords by made up words or words from Uygur language, instead they ended up having more loanwords from french. I would say still turkish has more arabic / persian loanwords than azerbaijani. On the other hand azerbaijani gained a lot of russian words during Russian Empire and Soviet Union, especially scientific words, as 99% of scientific words in azerbaijani basically russian translit in latin alphabet. In everyday life Azerbaijanis use a lot more russian words that official language.
There is another point, when we compare to languages - there is a dialect continuum from Baku to Istanbul. Baku and Istanbul dialects are the most different ones - if person speaks native Bakuvian dialect (not official Azerbaijani) I highly doubt Turkish people would understand even the subject, because dialect has many features that comes from Tat language (not Turkic language). On the other hand, if someone is azerbaijani from Iravan (present day Armenia), turkish people would understand him/her very easily. Because there isn't any dialect differences between Iravani dialect and Kars dialect. Then there is tiny difference between Kars and Erzurum when you go west and Ganja dialect and Iravani dialect when you go east, that way we can see dialect continuum.
Also I have to note that there is another dialect of Azerbaijani spoken in Iran, called South Azerbaijani dialect. Officially there language is called Turkish, because they didn't have soviet rule that resulted identity change. South Azerbaijani also have dialect continuum from west (Urmia dialect) to east (Khorasan dialect), first one sounds mostly turkish, second one sounds like turkmen language.