Most obvious differences: Bilinear or Trilinear texture filtering, smoother framerate
Next: Higher detailed 3D models, more polygons, better texture resolution, usually no noticeable popup effect and complex and more sophisticated ligthning.
And last: 3D engines look rock solid in Dreamcast, while there are some texture positioning problems (textures tend to shake in PS1 games) and some polygon glitches too.
Despite these differences, the PS1 does it very well with its very limited amount of RAM (2 MB) and video RAM (1 MB)
while Dreamcast with his 16 MB RAM and 8 MB video RAM can exhibit higher resolution textures easily.