Each of her three aspects covers a different part of her domain. In addition, each of the aspects comes with their own personality and proclivities, oftentimes directly counter to the others. Very little is known about her otherwise, as she is ever-changing and veiled, much like fate and luck themselves.
Fortuna is the aspect most focused on chance and luck. Landing a sword blow at just the right moment? Drawing just the right card at the right time? Fortuna’s hand can be seen in these actions, manipulating luck and chance to her wishes. However, Fortuna is a capricious aspect, and is as likely to help as to hinder. She favors the underdog, the long shot, but oftentimes breaks those rules with apparently little thought or plan. Due to her seemingly random nature, she wars most often with Tyche, and there are times where the two aspects of her are on opposite sides in a situation.
As much as Fortuna is random and aimless, Tyche is pattern and structure. The weaver of the Pattern of Fate, she governs the destiny of all, making sure that none escape the fate woven for them. She is very hands-on, pushing and pulling to ensure things go the way the Pattern plans. She is certainly the least empathetic of the aspects. While not cruel or mean, she is very stern, and takes her job protecting the Pattern of Fate very seriously. Woe to the person who escapes their fate, for her wrath can be terrible.
In the middle stands Tymora, the balanced aspect. She is the one to use both luck and fate to correct things, however she maintains a very careful balance between the two. Tymora is the judge, the balancer of the aspects. When Fortuna and Tyche war (and they do, often), Tymora is the aspect that makes the final decision. The reasons for her decisions are a complete mystery, but she stands uncontested as the final arbiter in all situations. To that end, it is Tymora that powers many divination requests, since she can continue to balance the other two aspects by giving mortals the information she wills.
It does not appear that any of the aspects care that someone would worship another. In that regard, they present as a singular entity, one that simply responds to multiple names. Only her worshippers can see the differences in the aspects. In addition, worship of any of the aspects does not guarantee protection or reward. She expects that her worshippers will do her work, but just as chance is fickle and fate is inevitable, they too will end up on the wrong side of the Three-Faced Goddess at some point.