From what we saw on screen, I have always gathered that the Galaxy-class was equivalent to the RN's Admiral-class battlecruiser (HMS Hood). They are both large and well equipped as flagships and they represent the pinnacle of each generation, however, they had fatal flaws that were exploited fatally in battle.
The Galaxy-class was designed and built during a turbulent time in UFP history, difficulties with the Klingons, the Cardassian Wars, the Tzenkethi War, and the Galen Border Conflicts were fought in the 2340s and 2350s, yet, they were nearly ended or in the process of ending in early 2360s, and so the Enterprise and her sisters were converted to a more peaceful role, the explorer.
I see the Sovereigns as the UFP equivalent to the USN Iowa-class battleship. They are fast, they are able to take a pounding, and can deliver one too. They were designed in the mid-2360s and they got to see the rise of the Romulans, the Borg, aggressive Klingon factions, and the Maquis insurgency.