Tai Mantis is an organization of Mantis instructors. The proper name of the organization is the Tai Mantis Kung Fu Association. Notable instructors are Paul Eng, Kam Yuen, Don Newton, Rob Moses and Manuel Marquez. Please forgive me if I left anyone out.
Taiji Mantis is an actual style of mantis that was founded by Sun Yuan Chang, student of Liang Xuexiang. It has since divided into different lines/families.
Huge question bro! LOL!
All the families that descended from Liang Xuexiang (Meihua, Taiji and Taiji Meihua TLQ) are very similar. Each would recognize the forms of the other that they hold in common.
Mainland Qixing TLQ has a very similar "feel" as the Greater Meihua Lines mentioned above, but the forms are a bit different.
HK Qixing TLQ is a whole 'nother animal.
It uses smaller circles and most would say is a bit more direct. The forms play very differently than some Mainland styles.
Please realize the above are generalizations, but I have no desire to write a thesis on the subject. LOL!