Israel was extremely tactically important in the ancient world. Without airplanes, domesticated camels, or deep sea travel, it's the only way to get between Africa (read: Egypt) and the rest of the world. Throughout recorded history empires have fought over control of that passage. Even by the Roman empire, when sea travel was more possible, it was still often easier to march armies across land than to transport them on ships, depending on how large they were and where they were coming from. Galilee also has a lot of arable land. It's not the breadbasket that Egypt is, so most conflict over the land was about controlling the territory and the passages to Egypt and points east, but the agriculture played a role, too.

During the first century CE, the region "Israel" was divided into multiple units. The ones we hear about the most are Judea, Galilee, and Samaria, but there was also Idumea, the Decapolis, and other districts in southern Syria. Collectively, they'd been fought over throughout the Hellenistic period by the Seleucid and Ptolemaic kings. In the 2nd century BCE, Judas Maccabeus was able to drive the Seleucids out of the region, resulting in roughly a century of sovereign rule. By the mid 1st century BCE, the Hasmonean dynasty was fracturing. This corresponded with extended Roman influence in the east, and in the 60s Pompey Magnus entered a Hasmonean civil war and established the region as a client kingdom of Rome under Herod the Great. After Herod's death his dynasty fractured, and Rome responded by making Judea proper (the southern region) into a directly controlled province ofRome, and dividing the other regions between his sons and their descendants, who acted as client kings. That's the simple version--Hasmonean and Herodian history are a saga worth George R. R. Martin, and the war that involved Vespasian and Titus was, in part, the saga's last flare-up.

The instability in Israel was a logistical and ideological problem for the Roman empire. Rome's takeover of the east had come relatively peacefully. Achaia (Greece) had allied with Rome for help against Macedonia, and regions in Asia (modern Turkey) and Syria had been fighting Seleucids and Mithridates. There were definitely wars, but Rome was able to cast all of them as the assistance of allies, never mind that they always ended with Rome taking administrative control or setting up client kings over the contested regions. The same thing happened in Israel (Pompey was invited), but the difference was that, after a century or two, most areas in Asia and Syria adapted to Roman rule without much organized resistance, whereas Israelite ideology, especially following the Seleucids, enshrined sovereign rule. Compare all this to Gaul and Britain, which had hostile borders and guerilla warfare.

Basically, Rome had enough to deal with in the north. If Judea successfully rebelled and took the other tetrarchies with it, that would not only leave a gap in Roman land and sea control between Syria and Egypt, it would potentially encourage pressure from eastern empires and/or other rebellions in Syria and Asia. So when the Roman governor of Judea was killed and the governor of Syria wasn't able to get things under control, Rome sent in Vespasian to deal with it. He was very well proven in war, but a bit of a political persona non grata at the time, so hopefully someone with more expertise in imperial politics will be able to illuminate why Vespasian specifically.

It's hard to overstate how seriously Rome took the rebellion in Judea. They burned Jerusalem to the ground (which Josephus suggests might have been an accident or set by the rebels) and then pulled down the stone walls that didn't burn (not an accident). That was actually fairly rare in Roman warfare: why knock down a whole city just so you have to pay to build it again? The three major destructive sacks--Carthage, Corinth, and Jerusalem--are all about making an example and flexing Rome's might. After Titus finished putting the rebellion down, he and Vespasian put on a massive triumphal parade in Rome, in which they displayed and publicly executed Simon bar Giora, one of the major leaders. They also minted Iudaea Capta coins (Wikipedia has a good article, with pictures) around the empire for 25 years. Plus the Arch of Titus in Rome commemorates the sack of Jerusalem. It was, in other words, a major tool of propaganda to promote the Flavian emperors and discourage further rebellion on the edges of the empire.

Some reading:

Goodman, Martin. The Ruling Class of Judaea: The Origins of the Jewish Revolt against Rome, A.D. 66-70. New York: Cambridge University Press, 1987.

Mason, Steve. A History of the Jewish War: A.D. 66–74. Cambridge: Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781139020718.

Vespasian wasn't "on the eve" of becoming Emperor when he went to Judea in 66, and he certainly wasn't Nero's heir. He was viewed as a political outsider with no real power base (he'd spent most of the last decade in the political wilderness) , and therefore someone safe and unassuming that Nero could trust with armies to put down a revolt. Vespasian's Imperial ambitions seem to have been hatched after he heard of Nero's assassination in the subsequent political turmoil (The Year of the Four Emperors, 69CE) , where he threw his hat into the ring after the acenscion of Vitellius (with the support of ex-supporters of Otho - the Emperor defeated/killed by Vitellius). Vespasian himself didn't even make it to Rome until after the civil war was decided - he was proclaimed as Emperor in Egypt (The key to Rome's grain supply), and it was his ally (Marcus Antonius Primus) who actaully led the army that took Rome.

Regardless, it's often common for a Roman heir to be involved in military operations to gain experience/popularity with the army. Tiberius was often in charge of the field armies during Augustus' reign, and Germanicus during Tiberius' (before his untimely death), indeed Titus (Vespasian's son) continued on in Judaea, and was actually the person who took Jerusalem after Vespasian had departed to go become Emperor in Rome.

As for the importance of Judea - it's particularly important because it borders two incredibly important provinces - Egypt, and Syria, and more importantly, is close to the Parthian Empire. After the destruction of the Hellenistic States in the Eastern Mediterranean, the Parthians were the only real state that could challenge Rome, and the Syrian border was heavily fortified. A strongly anti-Roman Jewish state in Judea could be incredibly problematic for the Roman position in the Eastern Mediterranean, and therefore absolutely something the Romans would have to deal with - especially as this revolt is less than half a decade after the end of the latest Roman-Parthian War over Armenia.

The Romans also have a fairly brutal approach to any sort of rebellion, other rebellions around the Judean Revolt (Boudicca's Revolt in Britannia 60 CE, or the Batavian Revolt in Germania Inferior in 69-70CE), were also pretty heavily crushed. From a Roman General's perspective, any sort of action is incredibly valuable to advancing a political career, and while the whole Triumph of the late republic was long gone (Due to Augustus limiting triumphs to only the emperor himself), it was an important way of self-enrichment, and still important for getting high political office, especially a lucrative governorship, through which you can become rich.