The Justice League/Authority analogy is a good one. The players are extremely powerful, but they have their weaknesses...and one of the biggest is that even though they are immensely powerful, their friends, family, countrymen, cohorts, fellow worshippers and innocent bystanders are NOT. Being able to crush an ultraloth like a bug isn't as effective as it might be if the bad guys send noname demons to harry the PCs relationships.
Another problem of Epic Spellcasting and the ELH in general is that it doesn't FEEL Epic, as was mentioned. Only one of my players has stayed single class in 4 years and 24 levels...and he looks at the list of feats and powers...and shrugs. The Epic Cleric looks at many of the balanced abilities and says "Ooooh. I can have a Turn Dead Aura on all the time to affect very weak undead. Like I'm using my 31 daily turnings (16 Greater Turnings) as it is, now. YAWN."
I haven't read Exalted, but from what I hear, it certainly projects an Epic feel. I look at Hero, Legend of Zu and House of Flying Daggers as examples of Epic Level Characters flexing their stuff. The character of Gatsu from the manga Beserk! is an example, IMHO of an Epic Level fighter (at least, eventually).
The ELH feels like a half-hearted attempt, sometimes. It's balanced, but seems to be missing a certain pizzazz.
On the other hand, the scope of adventures that I can offer at Epic Levels is dramatically more interesting and just as fun as low level adventures. Repelling a Githyanki Invasion of the Prime, stopping an evil overlord from conquering the underdark, throwing down an evil demigod, reclaiming an ancient city from the Demon Prince Orcus, himself; travelling to the Far Realms to rescue the oldest living dragons from.....something.
A part of Epic Level play, for me, is saying "Why not?"