What is different about Hard Mode?

The new Champion type is called Eternal, and are tinted white (like Eternal flies sprite). They have increased health and their attacks are more powerful and faster. They also shoot more bullets than usual, and even homing bullets.
As far as I can see the Eternal version is exclusive of Hard Mode, they can be both bosses and random enemies. If you defeat an Eternal Boss in addition to the usual pedestal item you get an Eternal Heart as reward.

Here's further info I've found on Reddit.

• If you roll The Dice it sometimes makes no item appear at all.

• Some Eternal bosses, like Gurdy Jr., now track you when they charges and they also have regenerating health.

• Eternal Peep has now more than 2 wandering (and annoying) eyes.

• Mom's Heart and It Lives have earned the ability to move.

• Bombs usually become Troll Bombs when you pick them up (as you wrote).

• Added one new trinket called Game Squid: its effect is making each X'th tear have Gish's slow effect. It's basically a trinket version of Ball of Tar without the creep.

• There should be a new type of room, Eternal Arena:

  It works the same way as a boss challenge room, however it takes items from the angel room pool and also spawns 2 hearts when you pick the item up. Also the bosses are all the diabolical white alts. Don't know how much health or w/e you need to get in though.

As a side note, the amount of bullets some of the Eternal enemies fire is enough to slow the game down dramatically. The old good Flash.