The source for Gibbon's oft-reproduced quote is the late and unreliable Historia Augusta, which says:

"[Gordian II] was very fond of women; indeed, it is said that he had twenty-two concubines decreed him, from all of whom he left three or four children apiece." (Three Gordians 19)

We have no idea whether Gordian actually had anything like this many illegitimate children. More pertinently to your question, we have no idea what happened to them. Decades later, the emperor Claudius Gothicus (r. 268-70) was rumored to be a son of Gordian (Epitome De Caesaribus 34). This, however, is almost certainly false.

Roman emperors seem to have had surprisingly few illegitimate children. Several early emperors (notably Vespasian, Antoninus Pius, and Marcus Aurelius) kept concubines, but no children are reported from these relationships. Since ancient contraceptive methods were unreliable at best, we need not assume that none existed. Some illegitimate children may have been exposed (this, reputedly, was the fate of a child Claudius' wife conceived with a freedman, and of the illicit offspring of Augustus' daughter Julia). Or it may simply be that historians were uninterested in illegitimate children.

In the later Empire, a few bastards were worrisome enough to be forcibly removed from the political stage. Galerius had an illegitimate son named Candidianus, whom Galerius' rival Licinius took the trouble to kill. Licinius himself had a bastard whom Constantine reduced to slave status.

At least in stable periods, illegitimate children were not a threat to the succession, since they could not inherit the throne. And they might - to judge from the career of one of the few imperial bastards we know anything about - prove useful, if sometimes dangerous, pawns. John Athalarichos was the illegitimate son of the emperor Heraclius (r. 610-41). He was apparently a (rather low-ranking) member of the imperial court, expendable enough to be sent as a hostage to the Avars. Late in his father's reign, in the midst of the chaos that accompanied the Arab invasions of Syria and Palestine, he was the figurehead of a plot to replace his father. The plot was discovered, John's nose and hands were amputated, and the imperial bastard was sent to live out the rest of his life on a small island.

Regarding inheritance and fratricide. As illegitimate children, that includes the results of relations either concubinatus or stuprum, which means the relation is improper and sometimes means criminal*,* rarely could inherit their fathers estates and so on, this would be pointless. There are many stories of fratricide in roman history starting with Romulus and Remus but probably most theatrical Caracalla and Geta. The main difference comes from what the mother is. If she is of (roughly) equal social class to the man, but not his wife, they might well be included in the will and considered full heirs. The lower her status the less like legitimate children they are. In early imperial roman history there are not a lot of cases where this plays a huge role

This obviously is different all across roman history.

A very prominent moment when an illegitimate son became a political problem occurred when Elagabalus, a relatively distant relation to the assassinated emperor Caracalla, was declared, falsely according to Herodian, an illegitimate of the aforementioned emperor by his mother. As Caracalla had no other heirs, Elagabalus would have had a good shot a succeeding him, which he did. Of how much consequence this claim was is uncertain.

What you can hopefully gather from this is that inheritance was a complex matter for Romans and that there is no easy answer. However there is also no definite prescribed path with illegitimate children will likely take.

Here is an Article about it you could check out. Illegitimacy and Inheritance Disputes in the Late Roman Empire" in Inheritance, Law and Religions in the Ancient and Mediaeval Worlds ed. B. Caseau and S. Huebner (Paris 2014)

For the claim about Elagabalus check Herodian, V, V, 10.