H81: As with H61 and other HX1 predecessors, H81 is most at home in "living room PCs" and use-focused HTPCs]. You lose all advanced storage functionality, drop 2 of the SATA ports, drop to a single x16 PCI-e slot, lose 2xPCI-e 2.0 lanes, and lose one DIMM per channel (meaning you're restricted to 2 memory slots). There's no overclocking support and BIOS is effectively relegated to controlling your boot devices, seeing as there's not much else to do in there.[/b
We generally push users toward the H87 and Z87 chipsets for their higher build quality -- even if you're never going to have more than 2 sticks of RAM, 1 VGA, and 4 SATA devices, motherboard manufacturers generally cheap-out on the consumer-class products (HX1). I personally feel much more comfortable recommending H87 over H81, even if you're saving $10 or $20.
H81 has its place, it's just not for the average reader of this site -- mid-range- and high-end-equipped gamers.
If you enjoy playing with settings, tweaking, and getting your hands dirty in DIY-type modifications, then opt for Z87 and K-SKU CPUs.
If you don't care for any of the above and simply want to play games and leave BIOS alone for the duration of the system's life, opt for H87.
If you're building a function-limited HTPC or desktop machine, consider H81, but favor H87 for its higher build-quality.