There was the immediate and as expected angry reaction from the Russians and apology from Turkey. This is highlighted in the Wikipedia article referenced. My question is did shooting down the plane change the dynamic between Russia and Turkey in any way or did relations just return to business as usual afterwards?
The shooting event affected relations very strongly.
Even "What if the Russian stop the natural gas flow?" is asked in broadly Turkey. Some Turkish students are arrested in Russia with no reason. Import from Turkey for most products are forbidden in Russia. And touristic organizations are banned to make holiday trips to Turkey.
But after 15 July 2016 coup attempt in Turkey it was necessary to be closer to Russia for Turkey because Turkey knew that the coup is made by FETO (Fetullah Gulen's organization) and Fetullah Gulen lives in US and US doesn't want to give Fetullah Gulen to Turkey. For Turkey it's like a country doesn't give Usame bin Laden and not acceptable. In short, counter-US politics forced Turkey to get closer Russia. And Russia thought that "if I can grab Turkey from NATO it is very good for me". Turkey needed Russia for Syrian issue too.
For these reasons both countries act like they have no problem with each other.
Added references in Turkish:
The title translated: "The biggest trum card of the Russia is natural gas" http://www.hurriyet.com.tr/rusyanin-en-buyuk-kozu-dogalgaz-40018314
This one is about Turkish students detained in Russia: http://www.hurriyet.com.tr/rusyada-yurda-baskin-yapip-turk-ogrencileri-gozaltina-aldilar-40020136
This one is about import-ban which includes tomato, onion, cucumber, strawberry, orange, chicken, torism tours, http://www.bbc.com/turkce/haberler/2015/12/151201_rusya_yaptirim_liste
And after all those things why Turkey and Russia became ally again in 1 year? The answer can't have a "clue" because this is politics.
I can't demonstrate what Putin or Erdogan are planning to you but I've strong impression like what explained above.