Bluetooth modules: Bluegiga BLE112 and HM10
Here are the differences I could find:

HM-10:   power: 3.3v (only)
BLE-112: power: 2.0 - 3.6V -- better for coin cell operation, for example

HM-10:   28 x 13 x 2.3 mm
BLE-112: 18 x 12 x 2.3 mm  (smaller)

HM-10:   available only from China  ($6.05, e.g. www.fasttech.com)
BLE-112: available from US distributors (Digi-Key &amp; Mouser, $15.66)

HM-10:   must be controlled by AT commands (requires separate µC)
BLE-112: can run code directly on device (8051), or controlled via AT coimmands

HM-10:   GPIO interface
BLE-112: UART, USB host, SPI, I2C, PWM, 12-bit ADC, GPIO interfaces

HM-10:   max transmit power: +6 dBm (better)
BLE-112: max transmit power: +3 dBm

HM-10:   sleep current: 400 uA to 1.5 mA
BLE-112: sleep current: 0.4 uA (1000 x better)

HM-10:   speed: 48 Kbps
BLE-112: speed: 100 Kbps

HM-10:   range (line of sight): 100 meteres
BLE-112: range (line of sight): 150 meteres


The HM-10 comes out better on price and slightly higher transmit power.  The BLE-112 comes out better on all other measures.

The HM-10 document you linked to said the HM-10 had FCC and CE certification.  However when I tried to follow the links to find the FCC ID's, there was no information.  The FCC and CE certifications for the BLE-112 are on the Bluegiga site.

The BLE-113 is similar to the BLE-112, except it is 1/3 smaller (16 x 9 x 2.1), draws less power, and uses the CC2541 instead of the CC2540.  However the BLE-113 has slightly lower transmit power (max 0 dBm) and a reduced range of 100 meters.  The BLE-113 also deosn't have the USB interface (all the others are the same as the BLE-112).  At $13.45, the BLE-113 is slightly cheaper than the BLE-112.