Music
Sonic Heroes has good music. Shadow the Hedgehog has good music. When it comes down to it, though, one is slightly better.
Main Themes
Sonic Heroes
Sonic Heroes is confident and matches not just the game, but Team Hero itself.
I Am...All of Me
The song hits many points of the game and Shadow's character in an edgy tone that aligns with the game's tension.
I like "Sonic Heroes," but I've never been as attached to it as most Sonic fans. While it made the opening awesome, it's immediate "Sonic Heroes, Sonic Heroes," on the press start screen probably hurt it in my mind...
"I Am...All of Me" is one of my favorites. I like the way it sounds as well as the anger and confusion expressed in the vocals. The lyrics are spot-on. And the "Here we go buddy, here we go," helps break it up so that it's not too "over the edge."
What I'm Made Of
An amazing song that voices both Metal Sonic and Sonic's determination to fight against each other.
Never Turn Back
Beginning with a soft piano, the song erupts into the story of how far Shadow has come and his determination to put the past behind him.
I like both songs a lot. "What I'm Made Of" is impossible to argue against. It keeps the boss fight alive. "Never Turn Back" is a bit slower, because it's for the closing. But the lyrics and style are perfect for Shadow, and the beginning is a nice tribute for Maria. I have to call this one a tie.
Other Vocal Themes
There are still many vocal themes for each.
Heroes
We Can (Sonic)
Though it's done in the same spirit as "It Doesn't Matter," it's not quite as memorable and despite the lyrics, it sounds more "Sonic" than "Sonic, Tails, & Knuckles."
Follow Me (Rose)
It has an "Amy" feel to it but it does not stand out and seems overly girly/cheery.
This Machine (Dark)
The song is epic. It's kinda' defined as Omega's, but with the lyrics it applies to each.
Team Chaotix (Chaotix)
Catchy and humorous. "They're gonna' kick your--TEAM CHAOTIX." Applies to the detectives easily.
Shth
Almost Dead (Dark and Neutral Dark)
Often sighted as the "edgiest" Sonic song, "Almost Dead" is probably a little too over-the edge. Still, it's loud chorus is fitting. It uses life and death (which apply to Shadow because of his "death" in SA2) as an allusion to good and evil (which apply to Shadow's struggle through the game): "no more life, no more death--somewhere in between."
Still, should have just been for the Dark endings, as originally intended.
Waking Up (Neutral)
Waking Up is more upbeat and applies to both "android" Shadow as well as Omega. The similarities to "This Machine" are welcome (both were by Julien-K).
Chosen One (Neutral Hero)
Chosen One describes Shadow very well, voicing both his regrets and his determination to become someone better. With it's more depressed/saddened tone, it matches the neutral-hero endings.
All Hail Shadow (Hero)
The original "All Hail Shadow" is a bit rougher than the Crush 40 cover, but it's notable lyrics and confidence were enough to make it a returning theme.
The songs Who I Am, Broken, and All of Me are good as well but I'm not comparing them since they did not make it into the game.
Even though the songs in Heroes were alright, overall the Shadow the Hedgehog tracks were better and more relevant.
Stage Music
Heroes
To be honest, much of my distaste for early Heroes levels come from over-play. Seaside Hill has made almost as many appearances as Green Hill. GREEN HILL. Overall the levels have a good, fast tone such as Bullet Station, or something appropriate for the setting like Mystic Mansion. I especially like Hang Castle and how it plays backwards for when the switch makes the level upside-down.
Shth
Even though Westopolis has to be played every time a new story is started, it hasn't gotten old. It's fast tone and intensity helps keep the level fresh. Most of the time the tones are focused on speed and the panic of the situation. Notably, though, there are slower levels like Lost Impact, which is calm and a nice contrast to The Doom. I also like how the "digital" levels are given more of a electronic sound. I like the remixes such as The Ark (Lost Impact is a remix as well).
Both soundtracks are good, but since some of Heroes' can get repetitive, Shadow the Hedgehog is just a bit better.