One big impact was that the Roman praetorian guard was the highest ranking and most powerful part of the Roman military for much of the Empire's history. It had constant access to the imperial family, the highest pay, the most prestige and some of the lowest casualty rates in the empire. As such, mostly well off nobles populated the guard (alongside others who the family took a shining to, or just the generally exceptional individuals. The praetorian prefect, in particular, had extensive legal and administrative functions which made them a hybrid civil servant and military leader.
Thanks to this access to important figures and the natural bias of its members, the guard quickly and repeatedly became embroiled in intrigue and plots. More emperors than worth counting became Barracks Emperors, meaning Emperors who took the purple thanks to the army rather than the senate or from inheritance. Many were acclaimed specifically thanks to the praetorian guard, including Caligula, Otho, and Vespasian. The habit of the military 'raising up' Emperors is linked to the Praetorian guard's inclination to raise Emperors themselves.
Finally, the praetorian guard directly led or intervened in military battles or campaigns that the rest of the Roman military undertook. Notable examples include in revolts in the 1st century AD accompanying Germanicus, in the Year of the Four Emperors, and the Marcommanic Wars. However, the guard became less and less engaged in military affairs from the third century onwards - many prefects were recruited from the class of lawyers rather than experts in war. Diocletian began rapidly weakening the power of the prefecture, until they were eventually disbanded and wholesale reform of the army took place in the late Western Roman Empire's history.