In any case, Fireball does twice the damage over more than five times the area. It is absolutely devastating. There are encounters that almost wiped the group at level four that I could handily win with a single casting of Fireball now that I am level five. The enormous range is also something simply cannot ignore because being able to hit an enemy at 150 feet means I can rain destruction on them for a half dozen rounds before they can get to me, and that is assuming they have a nice flat stretch of ground to work with.
I also have access to Counterspell which lets me simply negate a spell entirely on my opponent's turn. This is a whole new kind of power, one which lets me totally dominate another caster in a battle where it is a single enemy caster as the threat.
It isn't just me that got better at level five though. Many melee characters get a second attack at level five, allowing them to do twice as much damage. That isn't as good as 'twice as much damage and hit five times as many targets' but it is still an immense improvement.
This all makes encounter design a bit nutty. When I think of the fights that were insanely dangerous over the last few sessions all of them would be halfway defeated by a single Fireball, and if the monsters get unlucky it could easily wipe them right out. When I look at the theory behind monsters though they don't assume that fifth level characters are that much more powerful at all. The XP and challenge rating system assumes relatively smooth power increases for the characters but the actual game doesn't work like that at all. It isn't as though you can just put twice as many monsters into encounters and call it done either - they all die to Fireball.
I think this crazy step up in power for our party is just an artifact of the design choices of 5e DnD. They chose to have bounded accuracy so your bonus to hit goes from +6 at level 1 to +13 at level 20. It doesn't go up much, so they can't inflate character power by increasing that. They could continually give big damage bonuses, but instead they chose to give more and more attacks and have the attacks do similar damage. I don't like that solution because the jump between one and two attacks is so enormous and because it leads to longer and longer turns as the game progresses. Having high level characters just drag out combats is not a good design.
They really ought to give people more damage instead. It would mean that you don't have the incredible power jumps that lead to level four being super hard and level five being a cakewalk, and it would prevent drag on high level combat. For example, let people add their stat bonus to their attacks (not with offhand attacks) an additional time at various levels, or double their weapon damage roll. That increases their power but it does it in small chunks and doesn't add clutter in terms of extra dice and added resolution time.
Perhaps the best solution is to simply start characters at level five. That way you avoid the ridiculously dangerous low level time and everyone has lots of dice to roll. You do get more predictability that way, and that is a good way to avoid some of the swinginess (and constant deaths or wipes) that occur in low level play.
Also I am so going to yell FIREBALL constantly now. It will be wonderful.