I assume it is common sense you should run a lower pressure in the front wheel versus the back. I weigh 175 lbs (79kg), I have tires with max 120 PSI (8.3 bar), what PSI should I run in the front and the back tire on my road bike? I use a 700x25 tire (25-622 ISO/ETRTO).
It depends on the terrain you ride on, what you find comfortable and so on, so you need to play with tire pressures to figure out what fits you. Several companies (e.g. Schwalbe) give recommendations, but they're only good to start -- you need to play with tire pressures to find whats good for you.
The number written on the sidewall of the tire is generally useless. For most riders, a correct pressure is significantly lower than whats written on the tire sidewall, and the tire sidewall rating does not take into account the strength of the rim (wheel) you're running, so blindly obeying it can be hazardous since the rim may take damage if you hit a bump at max pressure. A correctly inflated tire absorbs road hazards appropriately and avoids pinch flats while not having too much rolling resistance.
That being said, a rule of thumb is that the rear tire is about 10% higher pressure than the front tire. But you're completely free to ignore this and find something that's comfortable for you.
As everything on this life, it depends. Having a bit less of pressure on the front tires works great for most people (Typically in the range of 5-10% less pressure). Under normal movement most of the load from the bicycle goes towards the rear wheel (A 55 to 65 percent, depends on the frame geometry and load distribution), so it makes sense to put more pressure, but there could be a problem when your front tire pressure is too low to support sudden and heavy braking.
As described Here
During hard braking, the entire weight of the rider is on the front wheel. Now the front tire carries three times as much weight as it usually does. If this isn’t factored into the tire pressure, then the tire can collapse during hard braking: Suddenly, the sidewall no longer holds up the tire. The tire loses the ability to transmit the forces from the road to the bike – braking and steering are seriously impaired. It’s like riding with a flat tire.
That being said, on my MTB I always run the same pressure for both tires as my main purpose is to avoid hitting the rim directly with a rock no matter if front or rear, whereas on my road bike I tend to put 10 PSI less, usually 90 for the rear and 80 on the front (Weighing around 70KG plus bike and gear).
I have never experienced such tire deflection and lost of control as an adult since I always run my tires at relatively high pressures, but I remember experiencing that as a kid and is not pleasant at all.
Long story short, you can run the same pressure at the front for added security when braking hard and cornering (Crits, tight curves at hardpack single track...) and sacrifice a bit of comfort in the process.