I own Age of Wonders: Shadow Magic, and looking at HoMM's screenshots, it looks very similar in terms of gameplay, theme, etc. HoMM III was released in 1999, and Shadow Magic was released in 2003. And so:
Q: Are HoMM III and Shadow Magic very similar in terms of gameplay, theme, etc.?
If they are very similar, is Shadow Magic better ("new and improved")? If they aren't very similar, in what ways do they significantly differ?
There are many differenmces.
AoW:SM is FAR more tactically combat oriented and also has a much more 'Civ-like' aspect. Troops attacking a city for example in AoW do so on an isometric screen and can approach from the north, south, east, etc. and can be used in a variety of ways for different ends.
HoMM 2/3 is much more simplified in that combat takes place on a simple 2D 'right-to-left' hex-grid. Ranged units(for the most part) must be protected by melee-tanks and are no good at close ranged combat(except for Titans, Zealots), Flyers can fly over castle walls, etc.
Heroes in AoW are no where near as fun as in Heroes III. There are a few basic classes that amount to generic "Warriors, Thieves, wizards, etc.) whereas in Heroes III there is one might and one magic hero class for each faction(so for the "Rampart", which is nature/woodlands based, there are "Rangers"(might hero) and "Druids"(magic hero) for example).
HoMM 3 basically has a lot of balance issues that never got worked out. The magic system is incredibly simple compared to AoW(which is not a bad thing) as there is no spell research type elements but a good %30 of the spells are completely useless and another 20% that are completely overpowered(re:Town Portal).
Hereos 2 and 3 had, overall better factions whose units and heroes made sense within the genre of Heroic fantasy(with some exceptions, such as the notoriously unbalanced and nonsensical "Conflux" town that was rushed out the door due to some idiot teenagers at the old HoMM "Astral Wizard" fan site. It featured pixies/sprites as elementals(?!) for example). AoW had a bunch of units within factions that made no sense, thematically).
The graphics in Heroes 2 were the best of any of these three games with the colorful hand drawn 2D sprites.Heroes III was intentionally dulled down and a bit more bland looking. AoW's graphics were nice but, as with Heroes III, a bit bland. However the exploration factor in Heroes 2 and 3 was fantastic! uncovering the fog of war and finding sites, monster-guarded treasures and such would keep you up all night with the "one more turn..." factor. AoW...not so much(but not bad either).
The music in Heroes 2 is still unmatched to this day in ANY game! Heroes III was not bad but not quite as good.
Heroes III is overall my favorites of these three games but all three feature some things that you wish the other two games had and so all of them are worth the money and very good games.