I don't have anything to say about the difference between the two, but two explanations of the Ashkenazi version in which the order is significant:
Acharis Shalom (found in Otzar Mefarshei HaTefilla): Normally a doctor can either be a נאמן or a רחמן – he can follow through with the treatment or spare the pain. ה' is capable of being both a נאמן and a רחמן – He can carry out the treatment and spare the pain. First comes the treatment, the נאמן, and then the sparing the pain, the רחמן.
Aishel Avraham 1: The Gemara in Avoda Zara 55a says that before Hashem sends any suffering, He makes it swear that it will depart at the exact time that it is meant to end. Hashem is נאמן to set a boundary and keep the illness bound to that. Then, He is רחמן that through prayer He removes the suffering early.
1 I neglected to record in my notes which Aishel Avraham I quoted or where it was quoted from. If I come across it again, I'll update this.