Blank doesn't own the Georgie Dome so he was missing out on all the revenue the other owners were getting around the league because they owned their stadiums. So he made the mild threat of leaving with the Falcons unless he got his own new stadium.
This is partially true. Blank did want more revenue streams, but I do want to clarify one thing you said: Blank was never going to move the Falcons out of Atlanta. There was talk about the old GM Plant in Doraville (NE of Atlanta) being a possible site for the new Dome but he was never going to move. Keep in mind that Blank built his fortune in Atlanta (Home Depot), his kids go to school here and he's deeply connected in the city, so he wouldn't ever leave. He ended up getting his new stadium a few feet away from where the Dome currently sits.
Also he and the city did offer $300 million in upgrades, but for the city to host Super Bowls, Final Fours, World Cups, College Football Playoffs they needed a newer, more modern facility.
The Braves situation is a bit more strange. They announced they were moving to Cobb in Nov 2015 after they couldn't get the deal they wanted from the city. The Braves moved for a couple reasons 1. To be geographically closer to "their fanbase"

and to make more money on property that they owned. They didn't own Turner Field (it was owned by the city and renovated into a baseball stadium after the Olympics in 96). They hated the area around it (

to Peoplestown, Pittsburg, Mechanicsville) and never really developed that area which is in the hood. Someone already mentioned that there's no MARTA stop there either. So when Cobb came with a sweet heart deal plus tax breaks, plus ownership for the Braves of the property that they will now play on , they took the money and ran. SunTrust Park is just about done now and they've got all types of attractions, entertainment options and housing there. But in a nutshell that's why they both left