I have both an original Spinali bridge head, as well as the Tiger Corana leather bridge head...and have owned a Moosehead bridge head.
The manufacturer of the Moosehead basically stole the design from Spinali, because Spinali did not have the design patented. There are small differences. The Spinali bridge was produced stateside, in CA, whereas the Moosehead is produced in China. The type of plastic used in the Spinali was of better quality, and a bit thicker than the Moosehead. Also, Spinali paid a bit more attention to detail, as there are no obvious raised 'seams' in the rest points on the Spinali that can cause unintended dingers in a shaft. They also seem to fit onto a cue much snugger than the Moosehead.
Take the time with the Moosehead once you receive it, and take some sandpaper to the inside of those rest points and sand them down well...this will help you to avoid those annoying dingers that can occur. As to making it more stable, Tiger sells spare grommets for their Corona leather bridge head, and they may fit the Moosehead and provide a more stable fit, as I believe the rubber is a bit thicker than that found on the Moosehead.
I carry both my Spinali and my Corona...as sometimes one is just better in certain situations than the other. However, since most pool halls, bars and taverns now have at least one Moosehead....getting proficient at using it will be an advantage.
As a side note.....Spinali bridge heads are no longer being manufactured. They ceased production once the knock-offs began surfacing for half the cost of the original Spinalis. The originals were sold individually bagged and tagged, with a US flag decal to apply to the bridge head, and were clearly marked Spinali....which is the name of the cabinet makers located in Garden Grove, CA who invented it.