I am conviced such a desaster is a great start for a TL:
1. The principate failed. This gives a lot of room to invent something better. Even if this is hard to achieve considering the limited political opportunities in ancient times.
2. After Rome was enslaved, burned down and salted, the romans regognize, that the northern border is DANGEROUS AND IMPORTANT. This might lead to a different Grand Strategy of the romans, which is more focussed towards the North than the East.
And this would change everything in the next 400 years!
This actually raises a really good question.
The Monarchy failed, the Republic failed and now the Principate (the Autocracy) as failed. What will the Romans do now?
I imagine that the Eastern border would suffer, Armenia might fall to Parthia (OTL Parthia subdued them in 37 and it took the Romans 10 years to get it back), Syria will most definitely be raided but I don't see the Parthians capturing Antioch and Judea might be a powder barrel having only been added to the Empire recently.
The client states in Mauritania might break free but Thracia is too close to Damaltia so those will be kept in check.
The Lusitanian tribes might give it another go at rebellion. They were finally defeated in 24BC so that's years to recover from the losses and with Rome burned and with the Empire in confusion I can see them another war. Most likely the northern iberian tribes would also try to revolt.
But in the end it all goes to who is in command in Illyria. Paterculus says that command of Dalmatia, in 9AD, went to Marcus Aemilius Lepidus, with Tiberius in Pannonia, but Dio says that Germanicus commanded in Pannonia and Tiberius in Dalmatia.
The only consistency is that both agree that Tiberius was in Illyria in 9AD.
Now this matters.
Germanicus, while popular by both the people and the army, doesn't strikes me as the ambitious type, if he was he would had fought Augustus in elevating Tiberius to heir.
Marcus Lepidus was by all account smart and a good soldier, managing the feat that his legions in Iberia didn't riot after Augustus death while the ones in Germania and Pannonia did, but most important he could play politics.
Tiberius is a bit of a wild card. He has only returned to power 5 years ago and given the way he ruled he never had any real love for the office of Principes and he never showed any liking for politics, but given that he is in command of the greatest Roman force in the area does he have a choice?
Independently of who is in Illyria with Tiberius he will have to either kill him and take command of the army or join forces with him. Either way the Romans need the army out of there and into Northern Italy as fast as possible.
But even if Rome gets back what will they do?
Monarchy was distasteful for them, the Republic is dead and buried and the Autocracy failed in the most spectacular way.
Maybe some kind of camouflaged Monarchy mixed with old Republican tradition?
Two men elected by the Senate to act as life Consuls but with the administration being kept in the hands of the Senate and the Consuls being glorified life long Generals?
Anyway one thing is certain. When Rome gets back on his feat, and it will they are still too powerful and ambitious at this point, the Germans and the Gauls are getting the full Lusitanian/Illyrian treatment. Prepare for massacres, mass enslaving and brutal tactics.
When Romans felt threaten, and in this case was threatened and in a Roman sense violated, its when they are they are the most dangerous. When they finish their revenge, and it doesn't matters how much time it takes (they really know how to hold gruges with Carthage being the best example), Germania and Gaul will be the greatest salt fields of Europe.