The main difference is that H55 and H57 have onboard, integrated graphics. So they are ideal for people who do not need a discrete GPU (and who get a Clarkdale CPU to go with it).
However, they cannot run SLI/CF in x8,x8 mode - but the P55 can (note: not all P55 boards do, but the chipset is capable of it). For some people, that's a deal breaker right there. There are exceptions to every rule, and in this case I have found one H57 (but haven't noticed an H55) that can run CF in x8,x8 - but just to add to the confusion this H57 board can't run in x8, x8 mode with a Clarkdale chip, it can only do it with a Lynnfield chip (i5 750 or i7 8xx).
See this thread for more discussion along this route, it gets a little off topic in the middle but still may be useful.
Another difference is that on the H55/H57 the memory controller is moved off-die, still on the package, but off die. This may have been done because of the Clarkdale's integrated graphics because it means super fast memory access for the graphics, but it also means slower access for the CPU core.
P55 supports 2 more USB ports than the H55; P55 supports 8 lanes of PCIe 2.0 (at 2.5GT/s) vs the H55's 6 lanes;
Intel Chipset Comparison Chart- P55, Q57, H57, H55
Some additional information- There is also the Q57. The H and Q-series of chipsets feature what Intel calls its Flexible Display Interface (FDI). This is basically a link between the CPU and the chipset that passes along video output. H5x/Q57 motherboards will have a video out on board so you can use Clarkdale’s integrated graphics.
Support for Intel’s Rapid Storage Technology is probably the main reason you’ll want H57 over H55. This Technology is also available on the P55 and Q57 - so it's just the H55 that does not support it. The difference between H57 and Q57 boils down to security and management features. The H-series is for consumers, the Q-series is for corporate customers.
Here's where a lot of this information came from:
http://www.anandtech.com/show/2901
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