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?
Turkey and Russia may not see eye-to-eye on everything, such as Turkey effectively freezing the Syrian war, by occupying northern Syria to continue their fight against the Kurds, whom they consider terrorists, and transiting stolen Syrian oil. Or Turkey supplying Bayraktars to Ukraine, or Turkey supporting Azerbaijan...
On the other hand, there are some issues, where they can come to agreement. Trade between Russia and Turkey is quite abundant - a large amount of Turkish clothes, fruits, vegetables are imported by Russia. Russian tourists often chose Turkey as their destination for vacation.
Hard to admit, but this is arguably what a healthy geopolitical relationship should look like: bargaining and compromises, combining mutual interests, with the individual interests of one's own country, but no unilateral bullying, decision making, ultimatums, threats and blackmail.
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.
This is an old question, but I will say how it looked like from inside Russia over these few years.
First, Russia imposed some economic sanctions.
Second, Russia* bombed some turkish military pilots at a military base.
Third, Russia** shot down one turkish military helicopter.
Fourth, Russia saved*** Erdogan from being overthrown, got their due reparations and considered itself pretty much satisfied...
- by the resulting amount of active pro-US forces in Turkey
- by the resulting relations between Turkey and US
- in general
So now it's back to business as normal, maybe even a bit more so.
I am not sure this is how it happened, but this is sure how it looked like.
*Well, no, but you get the idea.
**Well, no again, but are you with me?