Phillips are used as it easier for automated tools or for an assembly line worker to use. The bit just pops into the screwhead. With slotted screws, it takes much more time to get the tool into position and the bit can easily slip out and ruin the screw. If your Phillips screw was extremely tight, then you should have sprayed some penetrating oil on it first, then be sure the head was oil free. Use of the proper number Phillips bit is important as well. They also make what is called a Pozi Drive Bit ... it has serrations at the very tip where it fits into the screw and it will grab it much better than a regular bit. Also if you used a screwdriver, the better ones are the ones with the Black Tips as they are harder.
I don't mean to come down too hard on you, but I think I need to correct some factual issues in your post. Please don't take this personally, I have made many of these mistakes myself.
Phillips are terrible IMO. They do auto align better than a flat-head, but they are designed so that the bit 'cams out' before anything strips, which is why you have to apply so much forward force to keep the bit seated. Even then it hardly does its job because camming out tends to damage the screw and the bit anyway. Torx, allen socket, square drive, Frearson, etc. are all far superior to philips IMO.
Wiki article on 'camming out'
The 'hammer' type impact driver someone else pictured is ideal for removing philips bits that are stuck because the force from the hammer strike will almost always completely counteract the 'cam out' force from the bit.
Wiki article on impact drivers
Drive bits that are black (typically black oxide coated) are NOT harder than other bits, quite the opposite in fact. They are generally designed for use in impact drivers and thus are intentionally NOT hardened, so they don't shatter. They will generally be leaps and bounds better than your run-of-the-mill shitty powder metal bits, but not because they are harder. They will be made of stronger, more ductile materials (i.e. no chrome and different heat treatment) and typically are reinforced in key areas. Same goes for impact sockets.
There are, of course, regular bits that are just painted black. They are still pretty terrible. You'll get what you pay for, so don't cheap out.
Pozidriv bits aren't designed to fit a philips screw, they are designed to fit a pozidriv bit. The two are very easy to confuse, but they are different. Pozidriv bits won't fit a phillips screw properly, and phillips bits won't fit a pozidriv screw properly. They should not be interchanged. Using one in the other will typically cause damage.
Wiki article on pozidriv bits (and other drive styles)
You probably meant to call it an anti-slip phillips bit, but that is a very different thing and thus has a different name. They can help a bit (rimshot) but a hammer-style impact tool is a much better solution for this problem.
Pic of anti-slip phillips bit:
As for removing a stuck screw, I agree that penetrating oil is good and making sure the screw head is clean is very important (helps reduce cam-out forces). As is using the right size bit, this is quite frequently a #3 phillips (usually the biggest bit/driver kicking around) on brake retaining screws.
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