In the clear, none of those pesky trees overhead, both the Garmin and Might Mouse II drove most of the satellite signal bars up higher on a III+ than did the attached III+ antenna. I say most, because the signal level of the birds close to the horizon dropped when the Mighty Mouse was connected. Therefore, the Garmin wins the close to the horizon signal comparison test.
The Mighty Mouse II comes with BNC and MCX adapters. The Garmin is supplied with BNC or MCX, not both. Garmin sells a BNC to MCX adapter for those of use who have more than on GPSR. Just be sure to order the Garmin antenna with the BNC connector and use the adapter with MCX equipped GPSR's. Therefore, both antennas are compatible with the Map76 and III+. I'll give the Mouse a slight advantage in this comparison.
When it comes to under the hat comfort, it's a no brainier, the Garmin has it. The Mouse has a built in magnet, which adds significantly to the weight. The Garmin's magnet is attached with two screws. The Garmin also has smoother edges than the Mouse. The detachable base plate on the Garmin allowed me to fasten a rectangular piece of plastic generously covered with Velcro. This prevents it from wandering around under my hat and adds some strain relief at the cable attachment point. With the Velcro on the base of the antenna, I could shave my head and glue on a patch of Velcro and would not need to wear a hat.
In the real world below a dense canopy of trees, the Garmin antenna under my hat drives a stronger signal into my old 12 XL than what the built in antenna provides, but there is more to getting a true position than just signal level. Satellite geometry, where the birds are located in the sky, and the distortion of the signals as they pass through the tree cover can muck up the reported position.
Where on your body you place the antenna has a significant impact on reception. My Vista, with only a small patch antenna, performs quite admirably under a thick canopy of green if it is not blocked by my body. Due to the Vista's small size it has spent time under my hat and typically calls my pack's shoulder strap home.
So, whatever antenna you have, keep it riding high.
The Mighty Guru of Reception has spoken.