The Saints and Falcons are very similar teams historically. They were formed only one year apart, the Falcons in 1966 and the Saints in 1967 respectively, as the first teams in the deep south (unless you're counting texas). Almost like these two were made to hate each other.
But they are even more similar in the sense that they have historically, and excuse my wording here, sucked major ass.
Between 1967 and 1999, the Falcons had a total record of 198-295-5, the Saints had a record of 192-301-5 in that same span.
In terms of postseason success the Saints started out much worse, as they wouldn't make the playoffs until 1987 and didn't win a postseason game until the year of our lord 2000. 32 years after their inaugural season. The Falcons on the other hand would achieve both in 1978. However the saints would quickly catch up since 2000 and now both are tied up in playoffs wins with 10 each.
The Saints have of course, won a Superbowl in 2009 (although it is forever stained with a certain scandal), while the falcons have made 2 and lost both, the more notable one of course being their appearance in the 1998 when the 14-2 Falcons would fall to the John Elway led broncos.
All of this literary diarrhea just to say, This rivalry is something very special. My dad is a lifelong Saints fan and while being a fan of the saints wasn't very easy back in the day, he said Falcons games were always a treat to look forward to. The all time series is now tied 54-54 and the Saints have a shot at taking the lead for the first time since their very first matchup in 1967, when the saints would win at home 27-24, and then immediately lose the next 9 games straight.
It was nice watching Brees/Brady shootouts for a year. Now all those games have is the Mike Evans vs. Marathon Lattimore grudge match.