Early enemies become more dangerous
A goblin has an AC of 15, which is much higher than the DMG-recommended AC for a foe of its CR. In exchange, it has extremely low HP. Let's consider a level 1 melee character. With 7.5 (1d8+3) damage from a longsword, the character will, on average, barely kill a goblin if they hit. If the goblin instead has 12 HP, it will take two hits (around 4 attacks) to down it, doubling the length of already-dangerous low-level combats.
Control becomes more powerful
The fighter takes up to twice as long to kill an enemy with maximized HP. The wizard casts hypnotic pattern and it is exactly as effective as it was against enemies with normal HP. In my experience, the game is already slightly in favor of full spellcasters, and maximizing HP would exacerbate the situation.
Certain spells are much less effective
As @DanielZastoupil pointed out, sleep directly affects HP, and thus is far less effective against maximized HP enemies. Against a room full of goblins with 7 HP, sleep has a 96% chance of affecting at least two of them, a 65% chance of affecting at least three, and a 17% chance of affecting at least four. If a goblin's HP is 12, the chances drop to 42%, 0.4%, and 0% for at least two, three, and four, respectively. In my experience, sleep is a large part of low-level bard, sorcerer, and wizard effectiveness.