GTA VC is a completely new game compared to GTA III.
Please check out GTA FAQ (they are all over this newsgroup).
One of FAQ Posts (actually you can find it using groups.google.com search)
has complete list of all GTA versions.

Q: What's the naming and numbering convention of the GTA game series?
A: GTA (original; so we call it GTA1 but that's not it's real name)
  GTA London 69 (add-on pack to GTA1)
  GTA 2
  GTA III (first fully 3D version)
  GTA Vice City
  GTA Sin City (rumoured, allegedly announced as the next release)
  GTA 4 (many hope as a multi-player version?)

  GTA VC is NOT GTA4, neither will Sin City be. Some like to call it
  GTA 3.5 but they are wrong. The last variant of GTA3 will be the
  last one built using the current RenderWare game engine, suitable
  for PS2 and contemporary PCs.

  As for GTA 4, we should probably expect a radical evolution. It may
  be specific for the next generation PS3 and whatever the next 2-3
  years of PC evolution will bring in terms of raw computing power.

  No doubt Rockstar games are pondering what they can do to seriously
  up the ante yet again before someone else does. Let's hope they can
  usher in a new revolution.

J+P wrote...

as mentioned is other posts in this thread... they are completely separate
games

however... if you're trying to decide which one to buy... i'd recommend vice
city

as far as technical differences, they are essentially the same... they both
have the same engine, same graphics, same options, etc.

there have been a few new features added for vice city... including bikes,
helicopters, a plane that you can actually fly, better hideout locations,
and an 80's theme that everyone seems to love around here (i'm a couple
years too young to care, but it doesn't bug me either)

the story line is a little better in vice city... mostly because in gta3
you're working for a boss, then kill him, then another boss, kill him too,
etc... in vice city... you get sent down there to start up a drug operation
(something like that anyway) and instead you gradually take over the city
and...... nevermind

also... you get rewarded for getting 100% in vice city... this increases the
replay value a bit before starting a new game from scratch... you get
absolutely nothing for 100% in gta3

and as far as cheats go... the codes are different between the two games...
but the effects are generally the same

either way... they've both received a perfect score on the game review page
of my little personal site... so you won't be disappointed with either of
them

or on the other hand... just get both :)

- chris
alpamsfq - 417A