I ordered a replacement for one of the old Schwalbe Marathon 700x28C tires on my ancient road bike, and it turns out that "new" Marathons of that size are meant to be inflated to 80-110psi despite that the older ones I have are rated to 80psi... so, now my front tire has 37.5% greater pressure than my rear tire. Principally, tires do not need to match, but will this significant difference in "hardness" affect the handling/fragility of my bike in any way?
The markings on the tires for the pressures can be essentially ignored. They're a combination of marketing and legal departments coming up with essentially arbitrary numbers.
Find a set of pressures that works for you so the tires are properly inflated -- it should prevent pinch flats, but keep rolling resistance low and absorb road hazards and irregularities to some extent. The tire will deflect a bit when properly inflated. There are charts which you can use as rough guidelines, but you'll need to play with it.
If the tire is over or under inflated, it can damage the wheel and/or compromise handling and/or lead to flats. Sheldon Brown has a good explanation.
Often, you want to run different tire pressures (for example, the rear tire has more load, so you may want to go 10% higher in pressure or whatever, even if the tires are of the same size). If you're running different sized tires (which is not uncommon, either for lower weight, better handling or comfort; your front and rear clearances for tires are often different), you'll usually be running different pressures.
Chances are the new tires you ordered are essentially the same as the old ones, so you should start with the old pressure combinations you were using, and make adjustments from there.