The Second Veientine War (437-434 or 428-425 B.C.) was fought for control of the crossing over the Tiber at Fidenae, five miles upstream from Rome. At the end of the First Veientine War Veii had probably retained control of Fidenae (at this point Fidenae remained independent, but dominated by one or the other of its more powerful neighbours), but at some point between the wars the Romans appear to have regained the upper hand. It was used as a military base during a war against the Sabines, and a number of Romans settled in the city.
The second war began when Fidenae shifted allegiance from Rome to Lars Tolumnius, king of Veii. Four Roman ambassadors were sent to Fidenae to try and find out why this had happened. The Fidenates send messengers to Veii to ask Tolumnius what to do, and were ordered to murder the ambassadors (Livy records but dismisses a tradition in which the deaths were the result of a misunderstanding that came about because Tolumnius was distracted by a good role in a dice game).
Livy provides a clear narrative of the war. The first battle was fought on the Roman side of the River Anio, and was won by the consul L. Sergius Fidenas, although at a heavy cost. After this expensive victory Mamercus Aemilius was proclaimed dictator. The Dictator was able to force the Veientines to retreat back across the Anio, but only into the hills between Fidenae and the Anio. Lars Tolumnius held that position until the arrival of his allies from Falerii, while the Romans took up a position in the angle between the Anio and the Tiber, within sight of the city of Rome.
In the resulting fight (battle of the Anio, 437 or 428 B.C.) Lars Tolumnius was killed in single combat by Aulus Cornelius Cossus, ending an Etruscan cavalry fight-back and giving the Romans victory.
In the following year the Romans led armies into enemy territory, but no battles were fought. An epidemic at Rome prevented any further action in that year, and continued into the next. This encouraged the Fidenates to raid Roman territory. A combined Fidenate and Veientine army then crossed the Anio, and appeared outside the Colline Gate of Rome. Once again a Dictator was appointed, this time Q. Servilius. An army was scraped together, but a battle outside the gates of Rome was avoided when the Etruscans began to withdraw. Servilius followed them, winning a victory close to Nomentum (435 or 426 B.C.).
After this defeat the Etruscans withdrew into Fidenae. The Romans followed, and besieged the city, which fell after the Romans dug a tunnel into the citadel. Veii was now left standing alone. No help came from the other cities of Etruria, and a truce was agreed that probably ended the war.
The two different possible dates for this war come from a political controversy during the reign of the Emperor Augustus, at which time the linen corslet taken from Lars Tolumnius still existed. If Cornelius Cossus was military tribune when the single combat took place, as recorded in Livy, then the earlier date is true. If Cossus was consul, the view preferred by Augustus, then the later date is true (when Augustus viewed the spoils they were inscribed as having been deposited by A. Cornelius Cossus, Consul, but it is possible that Cossus himself added the last word after he became consul).
The situation is further confused by Livy, who records another siege and fall of Fidanae in 426 B.C. This could be true - Fidanae had changed hands several times already, so a rebellion nine years after the city fell to Rome in 435 B.C. is possible, or it could simply be that the same event entered the Roman historical tradition twice. Even if we take the second view, this doesn't held decide on a date for the war, as the real fall of Fidenae could just as easily have happened in either year.
Livy's own account offers a third possibility - a single war that lasted from 437 until 425 B.C with a truce after the first fall of Fidenae. War was renewed under Caius Servilius Ahala and L. Papirius Mugilanus (427 B.C.), by which time Fidenae was once again hostile. The Romans suffered a defeat near Veii caused by a divided command, and appointed Mamercus Aemilius as Dictator yet again. The Allies moved their base to Fidenae, the Romans formed another army outside the Colline Gate, and in 426 B.C. Fidenae was once again captured. This does look very much like a mistaken repeat of the events of the 430s, with many of the same people performing similar roles, although that could actually have been the case, with the Roman people calling on the victorious commanders of 437-435 B.C. after an unexpected defeat.
In either case the war ended with the Romans in control in Fidenae and a truce in place between Rome and Veii (according to Livy this was an eighteen year long truce, agreed in 425 B.C., which would expire in 407 B.C., two years before the start of the Third Veientine War.