Differences game mechanic wise:
HC1 there are 3 stats (attack, hit points, lust) that you increase the base starting point of by playing, completing (by getting early game over or by making it to day 15), and then spending earned points on. During play you level up and buy perks which increase either attack, hit points, lust, stamina recovery, or gold drops. You buy or find craft (by finding or buying recipes and materials) weapons and accessories to give you stat boosts or special abilities (reduced corruption gain, reduced damage to hit points or lust, etc.).
HC2 there are 5 stats (str, agi, vit, int, con) which each have a derived secondary stat (attack, stamina regeneration, max hp, max lust, addiction resistance, corruption resistance). When you complete a run you still get points to spend, but instead of directly buying up the stats you buy a permanent increase to starting stat points, or starting gold. When you start a new run you have points to buy up your stats, but unlike HC1 you can spend those points in different ways each time you start over. Upon leveling you gain points to increase your stats, which increases the derived secondary. You also pick advantages / disadvantages at the beginning of each new run, instead of the older perk system. I believe there is still weapons and accessories, but I haven't played enough of HC2 to confirm that, other than I have seen the forge in the city.