As your googling probably told you, they were the older heavy infantry soldiers in the Roman legions.

Specifically, they were the older soldiers in the Roman army of the middle republic, which is to say the early third to the late second/early first century B.C. sometimes called the Polybian Legion. (Not because that was the name, but because the Greek historian Polybius gave us quite a detailed description of it.) This, incidentally, was the time when the Romans fought their most famous wars (the Punic Wars, the second of which was against Hannibal) and started to turn into the world power in earnest, expanding from a purely Italian state into one with dominion over parts of Spain, Africa, the Balkans, Greece, and Asia minor, as well as Sicily, Sardinia and Corsica.

The most important thing to bear in mind about the Roman army of this time is that it was a militia force of part-time, at best semi-professional soldiers. They were farmers' sons who would serve in the legions for a campaign before returning home to civilian life, until they were called upon again. At the beginning of this period they would only serve for a few months at a time, though by the end service could last multiple years as wars became lengthier and took place in more distant provinces.

The Roman militia army was composed of the citizenry under arms, and as such consisted of a cross-section of the population, divided by age and wealth. The very young and the poor served as light infantry skirmishers, the young men (Late teens, early 20s) of modest wealth served in the first battle-line as hastati heavy infantry, those of mature years (later 20s, early 30s) as principes heavy infantry in the second line, and those of mature years (later 30s, early 40s) as triarii in the third line. Meanwhile, the wealthy would serve as cavalry, since they were the only one able to afford horses and practice fighting from horseback.

I write in much more detail about the army of this period and how it would be raised in my post here

The most surprising thing about the triarii would be their equipment: whilst most of the Roman infantry at this time already used early versions of the famous Roman throwing spear, short sword and large shield, the triarii continued to use older-style hoplite equipment with a heavy thrusting spear as their main weapon. Being older and on average wealthier, they would also have been able to afford better armour. (Roman soldiers at this time had to bring their own equipment.) I write in much more detail on the equipment of the triarii and the reasons behind it in this thread here

As for how they differed from any old veteran soldier... well, one difference would be that they were not actually professional soldiers, but militia-men like the rest of the army. So whilst they would be older, they might not necessarily have seen that much military service. Still, the Roman republic was an aggressive, militaristic state in a world filled with aggressive, militaristic states, and the times when Rome was completely at peace (signified by the ceremonial closing of the doors of the temple of Janus in Rome) were so rare that it happened only twice in recorded Republican history. Most triarii would have seen quite a few campaigns.

Another difference would be the way they were used: as a final reserve, forming the third line well behind the rest of their army, kneeling behind their shields at the start of the battle so as to spare their strength. (They wouldn't be the youngest and most energetic men, after all.) The idea was that these older, steadier men would keep their heads and be able to turn the tide if panic broke out, hold back the enemy if the rest of the army had to flee, or be able to be thrown at the enemy at the decisive moment.

Roman writers sometimes joked about this, comparing them to onlookers at and spectators in the theatre, as Plautus does in one of his own plays. ("Come now, everyone sit on the periphery, just like the triarii,") But they were also proverbial as the last-ditch hope, the ones who would only come into play if the situation was truly dire: "It has come down to the triarii." Either way, these sayings seem to indicate that most of the time they wouldn't see much action.