For our PoD, let's say Livilla and Sejanus are caught sometime in early to mid 23 CE; the son of Tiberius outlives his life, eventually succeeding his father as emperor. How is history changed? (One possible
butterfly)
Well, you avoid a lot of Sejanus' reign of terror while Tiberius was on Capri, though you might create an earlier reign of terror for Tiberius. If Tiberius still more or less retreats from public life, Drusus is effectively in control much earlier than he is officially emperor. Though this isn't necessarily guaranteed, since Tiberius may feel a duty to stick around to train his son in the arts of statecraft.
Assuming the latter, this is already a major POD. A Tiberius more engaged in the affairs of Rome (rather than just directing treason trials from afar and arbitrarily imprisoning and executing Germanicus's family) will still not be much well liked by the Roman populace, but he also won't be loathed anywhere to the extent he was IOTL. If Tiberius isn't going on his treason trials thanks to influence from Drusus, Germanicus's family will remain in much greater shape. Drusus Caesar and Nero Julius Caesar Germanicus would be alive at Tiberius's death ITTL, and without Caligula, Julia Drusilla would remain married to Lucius Cassius Longinus (who in turn wouldn't have Caligula around to kill him). Depending on her own actions, Julia Livilla might survive to life to a ripe old age herself, though her apparent tendency to engage in plots might mean she still suffers a gruesome fate.
The Julio-Claudian dynasty went through a very rough period as far as dyanstic health is concerned during the latter half of Tiberius's reign, Caligula's reign, and the bloody end of Claudius's reign and then Nero's reign. So it's definitely in comparative great health in this scenario.
Now onto Drusus himself. Drusus was almost as popular as Germanicus. To give you an idea of the esteem Romans held them in (at least after they were both dead), Cassius Dio compared the two to the mythical twins Castor and Pollux. Drusus did apparently have a sort of temper, though so did Germanicus, and by all appearances, Drusus actually seems to have been much more competent and effective than Germanicus. As for what Drusus would do in power, he has less need for a military triumph than Caligula or Claudius did, as he has an extensive and successful military career already under his belt-he doesn't need a military victory to prove his legitimacy and boost his prestige. So it's entirely possible that he does not launch a campaign of conquest anywhere, though it is still probably likely (depending on how long he lives) that he may engage in some military or diplomatic conflict with Parthia over Syria. IOTL Parthia challenged Rome by establishing Tiridates on the throne of Armenia in 52 CE, leading to the Roman-Parthian War in 58. It's certainly possible Drusus would be alive to oversee that war, and might challenge Parthian aggressiveness earlier, though it's also possible that Parthia would be more cautious dealing with an experienced leader like Drusus than they were with Claudius and Nero.
If Drusus does launch any offensive campaigns in the west, there are two obvious options: Germania and Britannia. The settlement and diplomatic and trading ties set in place with Britannia by Caesar had largely broken down and so an expedition was certainly warranted, though this did not necessarily mean a war of conquest; a simple invasion to put things back into a Roman favored order, like those that would be launched in Germania periodically, might be all that is considered. The same might be true for Germania; there will certainly be excursions into Germania as happened fairly off and on during this period, but whether another attempt at conquest is attempted is entirely up to Drusus. Certainly the opportunity is there.
Assuming for sake of argument that Drusus dies at the ripe old age of 63 in 50 CE, there's a few options for succession that could make things interesting. There is of course Drusus's son Tiberius Gemellus, who would be 31. But there are also the children of Germanicus to contend with-Drusus Caesar, born in 7 CE and now 43 and Nero Julius Caesar, born in 6 CE and now 44, and of course the now 38 year old Caligula (assuming of course, none of these people die unexpectedly or are killed). This is really the first test of the pitfalls of the technically non-hereditary Julio-Claudian system created by Augustus, something we never really got to see IOTL thanks to either mass sudden death in Augustus's case, or mass interfamily murder afterwards leading to surprisingly swift and clean successions until the dynasty died out. It's a nice stage for a proxy war between the Julian and Claudian factions that had been sniping at each other since the early days of Augustus's reign, since Tiberius Gemellus has no Julian blood.
I'm honestly not sure how this would work out, and a lot depends on how Drusus handles his future succession before he dies. His problem by this point would be that all of them are established adults-he can't pull an Augustus and make one of them his immediate heir, adopting another as their successor. A lot depends on their personalities, so speculating is hard as not a lot is known about any of their personalities, thanks to their early deaths. It could get bloody very quickly, and it may just as well be over just as quick.
Whoever it is, they will likely face an immediate challenge from the Parthians (see above) with regards to Armenia, so they will be provided with a ready chance for establishing themselves with a military triumph.
I once read a biography of Tiberius by Robin Saeger, and now I'm convinced-even though Saeger never said it-that Tiberius might have suffered from Clinical Depression. He was dark, dour, moody, and possibly a Republican at heart; and stuck in a position he didn't really want.
I hope this tl makes his life a little better...
Tiberius certainly showed signs of it. He withdrew from public life often, and I'm not entirely sure he ever wanted to succeed Augustus.