Back in 3e, there was a "one and a half" strength bonus to damage when wielding 2-handed weapons. I thought that made sense back then, but I see no such thing in D&D 5th edition. I understand this must be for simplicity's sake, but perhaps there's something I'm missing? Is there anything else that a two-handed weapon can do besides make it impossible to use a shield?
In 3/3.5 the two-handed weapon damage was 1.5x strength mod to keep pace with two-weapon fighting. When doing TWF you added half strength mod to damage to your off-hand attack.
In 5e, TWF doesn't add your ability modifier to the bonus action attack (without the TWF fighting style), so the wonky math isn't needed in 5e.
5e Two-Handed Damage
To Answer your question, there is a "bonus" in the form of bigger dice. Two handed weapons rock the d10, d12, and 2d6 damage dice, while one handed weapons are stuck with d4, d6, and d8 damage die codes.
Weapons that can be used either one- or two-handed have the versatile property, which gives you a bigger damage die when used in both hands. Longswords, for example, do 1d8 damage in one hand and 1d10 with two hands. In addition, there are class features and feats that improve two-handed weapon damage. The great weapon fighting style lets you reroll 1s and 2s with two-handed attacks and the great weapon master feat lets you trade accuracy for damage with attacks with heavy weapons (all heavy weapons are also two-handed weapons).
Two-handed weapons have a higher base damage, and class features/feats modify that further.
Two handed weapons have higher damage, but not much. If you are serious about weapon attacks, you have a Fighting Style, so I will calculate with that.
A Longsword with Dueling does 6.5 damage, a Greatsword with GWF does 8.33. If you factor in 20 Str, it is only 16% difference. A shield is easily better in my opinion.
Before the errata Great Weapon Fighting Style applied to Superiority Dice and Divine Smite, but not any more. So the relative damage difference is small, and gets smaller with every damage source you gain, like Hunter's Mark or magic weapons.
However, two-handed weapons make it possible to use the Great Weapon Master feat.
If you do the math, you will see how powerful it is. Just make sure you do not take this feat early on, as it could decrease your DPR with low base hit chance, and without advantage.
For reach weapons, Polearm Master is the key feat. Without that they do the same damage as a Longsword in two hands, completely dissapointing.
If you don't or can't use that feat, a shield and a one-handed weapon is most likely better.
If you are a not a Paladin or Cleric*, and don't have the Warcaster feat, you can cast spells without dropping your weapon, as you can hold a two-handed weapon in one hand. If you have a shield and a sword, you have to drop your weapon to cast a spell, but with a two-handed weapon you can just hold it in one hand while casting. This mostly concerns Eldritch Knights, as most other spellcasters with weapons go with Dex and drop Str anyway.
*A bit more complicated than that, if you can use a Holy Symbol with the spell as a divine focus, and it is displayed on you Shield, you do not need an empty hand. So a multiclass Paladin/Sorcerer can cast Bless with a shield and a sword in hand, but not Fireball. And if your Bard picked up Bless as a Magic Secret, he still needs a free hand.