Before I even get started; Flickr can not stop Instagram at this point. Flickr can not beat Instagram in terms of hockey-stick-growth. Even with Instagram's recent policy changes Instagram is on a trajectory to hit the nearest star and Flickr nor Bruce Willis can stop them now. But, to succeed they do not need to win - they just need to capture as many Instagram-escapees as possible.
Flickr has long since been very good at a few things; sharing, licensing, and interoperability. It is one of the reasons Flickr was included in Anil Dash's The Web We Lost; Flickr's API is world-class and the entire Internet can benefit from its rich offerings.
Instagram being bought by Facebook was the first step in the wrong direction in the eyes of many web veterans. And there are more and more web veterans every single day as the web gets older. Web veterans are people that know better. Web veterans know that Facebook, Twitter, and Instagram aren't the Internet. In fact, they are the antithesis of the Internet. These companies do whatever they possibily can to pigeon hole people onto their websites for as many hours throughout the day as possible. The rest of the web, the real web, tries to solve a problem for people while playing nicely with every other service out there.
Flickr made the first big step in capitalizing on Instagram's move to Facebook last week when they debuted a brand-new iOS application that has gotten rave reviews from web newb and veteran alike.
However, Flickr is too expensive for people casually sharing a filtered photo from their mobile cameras now and then. Yes, you can use Flickr for free for up to 200 photos but I think just about anyone with a Flickr account would much prefer to have all of their photos available all of the time.
If Flickr were to change their model just slightly - one from a pro backup and catalog solution to one of sharing - they could easily win a ton of accounts that are falling off of the Facebook/Instagram table on a daily basis. Perhaps creating a cheaper account-type that costs, say $5 or $7 per year, would be enough for the web veterans (again, there are a lot of us) to completely jump ship from Instagram and pony up. This way, we would never have to worry about advertisements, creepy data collection, or wondering if our data will ever be trapped on someone else's servers. And believe me, this isn't something that is terrible difficult for Flickr to give a shot. They have everything they could possibly need already in place to do this.
Flickr, you've already made one step. Take the next step and bring us all back home again.