Introduction

The GameCube version of Sonic Adventure is praised by some but also criticized by many for being a so-called "bad" port of the original for Dreamcast. It has a lot of changes, sure, but most of those changes are improvements and additional content that more than make up for its shortcomings.

The following screenshots of the Dreamcast and GameCube versions of Sonic Adventure are from a twenty-minute video I made comparing the two versions. Both versions were played using their original consoles. No Emulators, no backwards compatibility. Furthermore, the Dreamcast screens are from a later copy of the game containing a revision (NA Version 1.1) that includes full Japanese and English voice tracks, some different camera angles, and bug fixes, among other things. Sonic Adventure DX for GameCube is based on the revised Dreamcast version and has most of these same improvements and more.

Why Sonic Adventure DX is better

Graphics: 

Since the GameCube is a more powerful console and SADX was released four years after the Dreamcast version (five years, if you count the fact that the Dreamcast version was released in Japan in 1998), the character models were enhanced with higher polygon counts, shadows were added, the draw distance was improved with less pop-up, water effects look better, a floor/ground was added in the Chaos 4 battle, and the game runs at 60 frames per second (which was what the Dreamcast version was originally going to run at). Other details were also added in certain areas.

This is what Sonic looks like. He, as well as the other characters, look somewhat blocky in the Dreamcast version. He looks rounder in the GameCube version due to the increased polygon count, more closely resembling his appearance in Sonic Adventure 2: Battle.

Various shadows were added to the GameCube version.

These are examples of the GameCube versions improved draw distance (you can see more in the distance). In the first two pics (taken in the first part of Emerald Coast just before the Whale area), you can't see either the building, bridge, or loop in the Dreamcast version but you can see them in the GameCube version. In the last one, look where Sonic's right hand is. You can't see those Dash Panels or Springs from here in the Dreamcast version, but you can in the GameCube version.

The water was noticeably improved in the GameCube version, looking distorted and wavy. In the Dreamcast version, the water effects are simplistic in comparison.

The Chaos 4 boss battle. Look underneath Chaos in the Dreamcast version; there's nothing. In the GameCube version, a floor/ground was added.

Outside of the Casino in Station Square. The Dreamcast version looks a bit generic here, while the GameCube version has a "Casino Ken" sign. 

This is the door in the Casino area of Station Square that leads to the Hotel. You can see that the textures inside don't match in the Dreamcast version. The floor has a red carpet in the GameCube version, both when you're outside and inside the Hotel.

Audio: 

Both the Dreamcast and GameCube have excellent audio capabilities and are comparable. However, GameCube discs have more space. As a result, due to the inclusion of both English and Japanese voice tracks, the audio is slightly muffled in the Dreamcast version. Most of the sound effects sound pretty similar in both versions, but the music and some of the sound effects sound clearer on the GameCube. The GameCube version also supports Dolby Pro Logic II surround sound. Unfortunately, the audio can't really be compared in a blog but I'll point out some of the areas where the GameCube version sounds better.

The whale section in the first area of Emerald Coast, and the fish jumping out of the water at the end of the second area of Emerald Coast. The splashing sound effects are pretty quiet and muffled in the Dreamcast version. In the GameCube version, these sound effects are loud and clear.

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If you stand by a door in the Dreamcast version, the door opening/closing sound effect plays over and over. This was corrected in the GameCube version, so that the sound effect only plays when the door is actually opening and closing (not while it is already open).

The monitors in the Station Square Chao Garden are silent in the Dreamcast version. This was corrected in the GameCube version.

The Egg Carrier doors are also silent in the Dreamcast version. Fixed in the GameCube version.

Extras: 

The big reasons why the GameCube version is the best. None of these features are in the Dreamcast version.

The GameCube version adds a Mission Mode, which has sixty tasks to complete. Two additional music tracks can also be heard in this mode when you start and complete a Mission.

The Mini Game Collection Mode contains every Sonic game released for the Game Gear. Literally all twelve of them, including Sonic Drift and Tails' Skypatrol which were originally only released in Japan. Games that include Versus Modes can still be played with a second player, and even the games that don't have a Versus Mode can be played with a second player. In addition, if you set the text language in the options menu to Japanese, you can play the Japanese versions of some of the Game Gear games. These games can be unlocked by collecting Emblems, as well as completing Missions in the Mission Mode. 

A Camera option was added to the pause menu in the GameCube version. You can choose the original "Auto Camera", or the added "Free Camera" option which can be moved around with the C-Stick. 

Also added to the pause menu is a Map option for viewing Maps of the Adventure Fields (Station Sqaure, Mystic Ruins, and Egg Carrier). It's pretty useful in the jungle area of Mystic Ruins.

Yet another addition to the pause menu. If you choose Quit, the Dreamcast version just goes back to the title screen and doesn't bother asking you if you really want to or not. The GameCube version asks you if you're sure you want to go back to the title screen, allowing you to go back to the pause menu if you made a mistake.

As mentioned above, the GameCube version has Dolby Pro Logic II surround sound. Also, for some reason, the Memory Card icon on the options menu isn't animated in the Dreamcast version but it is animated in the GameCube version.

The Dreamcast version doesn't allow you to skip cutscenes. Cutscenes can be skipped in the GameCube version.

The Station Square Chao Garden has several additions in the GameCube version, most of which are from Sonic Adventure 2: Battle. First, there are Action Windows telling you what button does what (the controls in the Chao Gardens are somewhat different than the levels). The Black Market is also included, allowing you to spend the Rings you've collected in the levels and buy many of the Chao items from Sonic Adventure 2: Battle. If you pick up a Chao, their Stats are shown. A Chao Board was added beside the door that leads to the Chao Stadium. The Chao Transporter is largely the same as in Sonic Adventure 2: Battle, but here you can also download a few mini games to a Game Boy Advance.  

Cream the Rabbit, who first appeared in Sonic Advance 2, makes several cameos in the GameCube version in both the Adventure Mode and Mission Mode. In the Adventure Mode, she can be seen flying around Station Square. In Mission Mode, depending on the mission, some Sonic Advance 2 signs appear. Chris Thorndyke from the Sonic X cartoon also makes a cameo.

Both versions have 130 Emblems to collect. Collecting them all is very difficult and time-consuming. But the GameCube version actually rewards you for collecting them. In addition to the Game Gear games, collecting all 130 Emblems in the GameCube version allows you to play as Metal Sonic in Sonic's levels in the Trial Mode.

Conclusion 

For whatever reason, the GameCube version of Sonic Adventure got poor ratings from critics compared to the original Dreamcast version. But as you can see, it makes several improvements and adds so many features that it might as well be the definitive version of the game. That being said, I can understand why some people are nostalgic and might prefer the Dreamcast version. It was one of the Dreamcast's launch titles and was a huge deal in 1999. The Dreamcast version is the original experience, while the GameCube version is the definitive experience. I'm glad I have both versions.

-MegaMan52