I'm re-watching the series now and just finished season four and am on episode two of season five so some things are fresh in my memory.
Carmela throwing Tony out was going to happen sooner or later. After finding the fingernail, smelling perfume on his clothes and all the other obvious evidence of cheating over the years the phone call from Irina pushed her too far. Part of it was that AJ answered the call and that brought Tony's affairs into their house, a long as it was at arms length Carmella might have been hurt, angry and jealous but it was not confronting her immediately. As it's been mentioned by various characters at different points in the show a woman or women on the side comes with the territory.
The other thing that I think had to do with the separation was Carmella was felling the 25 year itch. They married young and while Tony was out sowing his oats she was at home busy with the kids and making a home. With Meadow leaving and a sense of security her mind started to wander. Thoughts of a different man entered her mind. First it was fantasizing about Furio then taking a step further with Victor Musto. She knew that there would be severe repercussions if she had an affair while Tony still lived in the house so to avoid that and to help justify her actions (Catholic guilt) she needed Tony gone before testing the waters.
Unfortunately, Carmella realized that the life of a single, unemployed, middle aged woman with no work history (or ethic) is not easy. Especially when alimony is not going to be much. For physical and financial security reconciliation was her only option.