Here's some of the biggest changes I've noticed (with a few of the most impressive enemy additions):
LEVEL 1: This one is largely in enemy placement/numbers. No big deal.
LEVEL 2: Water sections have more bottomless water pits with more mermen and jellyfish. The second key is guarded by a number of fleamen, a knife-throwing lizard, and a zeldo, instead of one fleaman. You have to deal with an additional knife-throwing lizard while going up the stairs prior to this, too.
LEVEL 3: The opening section has no safety platform above the first jutting spike-spear and puts a barrier of red skeletons and a bone pillar to stop you from advancing without dealing with said spike-spear. Medusa heads are everywhere on the vertical sections with the jutting spike-spear gauntlets. Vertical sections have less "safe spots" and more flipping platforms/spikes working in conjunction with the platforming. The broken bridge section has a ton of red skeletons and one skeleton is placed with only a pixel-perfect margin for error (this one can be cheap).
LEVEL 4: You start out coming from a staircase above the knights and fleamen, and have to drop down while using the chandelier strategically. A red skeleton chases you as you deal with the bone pillars in the second room. The first vertical section now features more enemies, including medusa heads, on new sets of stairs that end in platforms that flip if you don't transition to the next set of stairs fast enough. If you take the tower route, there is only one tower (not two), making it a different climb. The collapsing bridge has many more enemies that come from all directions. The spider route has more knife-throwing lizards that team up with the rats.
LEVEL 5: The platforms of the platforming section after the second rolling eyeball launcher have been considerably narrowed, and medusa heads/floating eyeballs are added to the poltergeists and skeletons, forcing you to cleverly use the flipping wooden platforms to your defense. Pixel-perfect jumping is required even as you dodge/whip your foes. There is no crucifix/rosary to destroy enemies, including an additional floating eyeball, in the downward clock section. No extra life behind first locked door, only score items. Getting to the rewarding shortcut requires surviving a new platforming section that was once essentially solid (with collapsing platforms and bats still in place). The latter pendulum platforming section has different platforms and enemies involved.
LEVEL 6: Same as before, but Dracula takes more hits and his final form will REALLY give you a fight now. Better bring some axes.
Also of note: I pointed out a few things, but items are scattered differently in Hard Mode, too, and you lose more life when hit.
All of these changes make the progressive challenge of the levels more fitting and force you to really mind your use of sub-weapons at the right time. Bosses, more often than not, require you to learn their patterns, and not just whip away. It takes longer to learn, and longer to master. It can make a game that takes 30 minutes to an to hour to beat more like three hours to 3 1/2 hours long because of all the many things that can go wrong (player-caused). It's a more rewarding, complete experience, though, with nothing ever feeling remotely empty.
Now, I know that some Castlevania games have more enemies during a "second quest," but I don't recall any actually having variations on their physical platforms and layouts on any extra mode. I thought it was impressive that they took the time to make these changes, and I haven't heard many, if any, talk about it here.
PS: In case anyone is wondering, Easy Mode starts you off with 20-some hearts every level, damage rates are very forgiving, and Dracula only has two forms.