First and foremost, the show's usage of the term "Barbarian" is exaggerated and sometimes incorrect. While its ample meaning as "non-Roman/Greek" is Truth in Television, Romans would have never referred to Carthaginian citizens as barbarians, among other things because Carthage was a prosperous commercial empire with a very high culture and Greek influences just like Rome itself (the term was still theoretically fit for them, and Romans could have used it as an insult or to make a point during the Punic Wars, but definitely not as a casual tag). Similarly, Spartacus was a barbarian because he was born in Thrace, but it is unlikely Romans would have emphasized his barbarism as a sort of definitory trait, given that he had been a Roman soldier and gladiator for many years before his rebellion.
The barbarians were not freedom fighters against the evil, pro-slavery empire of Rome, as slavery was common virtually everywhere at the time. All barbarian tribes presented in this show practiced it, some of them even indulging in human sacrifices, ritual mutilations and other horrifying things even Romans were scandalized at (some of them rightfully, others not so much, but still).
Hannibal was not a freedom fighter as stated in the show, but just a military man motivated by a family feud against another nation. The war between Rome and Carthage can be best described as The Empire vs. a Mega-Corp, with both of them being pretty morally questionable by modern standards. Strangely, the show itself doesn't shy away from showing that Hannibal's oath against Rome was directly based on revenge, which turns it into a sort of inner contradiction.
Lusitanians didn't wear Gallic trousers as portrayed in the series, but short tunics down to their thighs. This is strange, as otherwise the show got several elements of their tribe surprisingly right (though, sadly, not their weapons).
Cumelios brands Hannibal in the arm with the burning tip of his falcata to test his bravery and mark their pact or some other cultural mumbo-jumbo. This ritual seems to have been made up for the series, as it is not recorded in any source.
All the Boii Gauls from the series are portrayed with full, short beards and buzzcut-esque haircuts. Historical Gauls are described to usually shave their chins and sport long moustaches, as well as flowing manes of hair. This is another weird point, as Cumelios and some Lusitanians (though not all) do have more or less correct hairstyles for their own culture.
The Iceni are shown to paint their faces, which there is no historical evidence of them doing.
Members of Attila's entourage wear what seem to be modern hoodies.
The Roman army was not composed by professional soldiers until 107 BC, despite what guests of the show claim in the episodes set before that year.
Contrary to what the series shows, Lusitanians weren't a huge part of Hannibal's army, but only a small part of a diverse Hispanic contingent. It also included Iberians (from the southern Iberian peninsula), Celtiberians (from the peninsula's core) and Balearics (from the Balearic Islands), as well as possibly other, minor tribes. In fact, according to a source, the Lusitanians were so few that they were bunched together with a Gallaeci contingent.
Neither Romans nor Lusitanians used bows and arrows at the time of the series.
Barbarians Rising follows the popular belief that the falcata was a Lusitanian weapon, despite it was actually an Iberian one and therefore limited to the Mediterranean coast. Lusitanians favored straight gladii, like those used by Celtiberians and later adopted by the Roman themselves. The series might have got away with it by leaving to interpretation that the falcatas were looted or bought from other regions of Hispania, but instead it shows the Lusitanians explicitly calling them "their fathers's swords".
Fairly universal throughout the series are cheap floppy leather breastplates and helmets on both Roman and non-Roman extras.
Hannibal wears a bizarre fusion of an Attic type helmet and a Late Roman ridge helmet. Putting aside the fact that it's clearly made of modern plastic, it also has a Chi-Rho emblazoned on the front, a Christian symbol that wouldn't come into use until centuries after Hannibal's death.
At one point Hannibal also wields what appears to be a replica of a Model 1816 French artillery short sword, ironically a weapon that was inspired by the Roman gladius.
Boudicca wears a leather dress which has been cut into scale shapes to resemble lorica squamata, while one of her lieutenants wears a late medieval gambeson.
Alaric, born in the 4th century, wears a very faithful reproduction of 1st century lorica hamata, which by his time had long since been replaced by more regularly constructed mail hauberks.
Hannibal and the Carthaginian elite were of Phoenician origin, which would make them look physically look vaguely like modern Israelis, with olive skin and Semitic traits. Meanwhile, the show portrays them with black actors, and some of the guests imply rather unsubtletly that the Punic Wars were somehow important for black people's history. In real life, there were probably few to no black people in the Carthaginian army at the time, as most of it was composed by other Phoenicians, Berbers and Caucasian mercenaries like Hispanics and Gauls.
The character of Cumelios seems to be based on Hannibal's Lusitanian commander according to Silius' Punica, as he even dies in Cannae the same way Silius tells about the historical guy. However, the latter is described as young instead of elderly and named Viriathus instead of Cumelios. (The name change can be justified, though, because viewers might have become confused at having two important characters with the same name.)
The show makes it look like the Scipio who intercepted Hannibal in Hispania and the one who fought him in Zama are one and the same. In reality, they were respectively father and son. They also imply that Scipio Sr. was present at the defeat at Cannae, when he was actually in Spain at the time.
Ditalcus wasn't a Lusitanian, but a Turdetanian deserter who joined Viriathus's party after it took off. Even if they had got it right, his tragic background (namely, that his entire tribe was butchered in a punitive action triggered by Viriathus) could not have happened in real life, as at the time Turdetania formed part of a rich Roman province.
The Lusitanians didn't ally with Hannibal in an Enemy Mine scenario against Rome. In real life, the Spaniards that joined Hannibal did it because they were either already Barcid vassals or rogue mercenaries seeking to make a buck, not because any of them gave a crap (yet) about the threat of Rome. In fact, Spanish mercenary parties also fought for Rome during the war and afterwards.
In the series, it is revealed that Hannibal had previously arranged his encounter with Magalus and the Boii Gauls. In real life, the Boii appeared upon Hannibal unexpectedly, so much that he had been thinking on changing his plans and only continued his travel as thought because Magalus convinced him to do so.
Magalus claims there are few Boii Gauls left, and only a few of them join Hannibal and Cumelios, implying the Romans killed the rest. In reality, Gauls (both Alpine and Cisalpine) composed the biggest reserves in Hannibal's army, as many of them hated Rome and joined him through his travel to Italy. Gaul was still mostly unconquered by Rome at the time and would remain so until Julius Caesar's time.
In real life, Viriathus took several years to kickstart his rebellion after the Massacre of the Lusitanians, not a single night. By the time it exploded, Galba had been replaced by Vetilius a long time before, so he never witnessed firsthand its effects. Vetilius was not an Arch-Enemy of Viriathus, either: the battle of Tribola was actually the first, only and last time they faced off, as it was at that very battle where Viriathus became famous and Vetilius died, all at the same time. The statement made in Barbarians Rising that Vetilius was fixated on Viriathus and carried a manhunt for him during many months is just plain wrong.
Nothing in history indicates Marcus Crassus paid the Sicilian pirates off. They apparently betrayed Spartacus on their own.
Boudicca did not get killed by a Roman soldier in battle as the show suggests. According to Tacitus, she took poison shortly after the battle.