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Imagine you are a level 5 Variant Human, selected a class with access to martial weapons and Extra Attack, and chose the Crossbow Expert and Sharpshooter feats.

You are able to attack two times a turn with a heavy crossbow as an action and three times with a hand crossbow as an action plus a bonus action. The damage you would do (assuming 16 Dex) with a heavy crossbow is 2×(1d10+3+10)=37 on average. The damage with a hand crossbow is 3×(1d6+3+10)=49.5 on average.

If you do not have a reliable use of your bonus action (e.g. Rogue's Cunning action, etc.), when would a heavy crossbow be superior to a hand crossbow?

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    \$\begingroup\$ Would you want answers to look at various values for Dex and hit chance, and also to account for critical hits? Also is there a reason you think the weapons should be equal? Do you want range to be considered? \$\endgroup\$ – Medix2 Mar 7 at 1:09
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    \$\begingroup\$ Both a heavy crossbow and a hand crossbow's chance to hit will scale equivalently with Dex. If anything, a hand crossbow gets stronger as Dex increases as you attack one additional time so get 1 more damage point than a heavy crossbow every time your Dexterity modifier increases. I would be interested to see how crits/crit fishing could impact things as a d10 would benefit from a crit much more than a d6. \$\endgroup\$ – ObeseMonkey Mar 7 at 1:13
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    \$\begingroup\$ Yes, as you've said, the number of attacks made means they won't scale identically with dex. That said, I assume you want answers to assume there is no other use for the bonus action when using a heavy crossbow? I think I'm just a bit confused on what the criteria for consideration are. Though perhaps they aren't necessary \$\endgroup\$ – Medix2 Mar 7 at 1:36
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    \$\begingroup\$ Yes, I was hoping to get answers that assume there is no other use for a bonus action. It is possible that the answer is "hand crossbows are better if you do not have a good use for your bonus action." How often you use your bonus action, I have found, varies greatly from class to class and subclass to subclass. A pure Champion fighter only uses his/her bonus action for Second Wind once per short rest whereas a Battlemaster fighter could potentially use his/her bonus action 4-6 times per short rest on manuevers. As some classes don't need their bonus action, this seems important to answer. \$\endgroup\$ – ObeseMonkey Mar 7 at 1:54
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See "All Together Now" at the bottom for the summary.

You've kind of answered your own question. Assuming that you're level five, you have Extra Attack (1), those two feats (and you're using the sharpshooter bonus), and a Dexterity of 16...

Hand Crossbow is going to be better in that scenario.

49.5 damage for three hits is just better than 37 damage for two hits.

However, there are some things that skew this one way or the other.

Factoring in hit-chance

If you're not using the "bonus" from the Sharpshooter feat, that brings you down to 17 damage (Heavy) vs 19.5 damage (hand). It's still in favor of hand crossbow, but when combined with the points below, it might tip the scale.

Factoring in critical hits

Hits are more or less likely to be critical hits, depending on how likely it is for you to hit in the first place.

If you need a high roll to hit, then it's likely that many of your hits are going to be critical hits. Expected critical damage is 28 (Heavy) and 26.5 (Hand), if you're not using the sharpshooter bonus (48 heavy and 56.5 hand, with the bonus).

This is especially true for Champion Fighter's expanded critical range.

Keep in mind that the half-orc trait, brutal critical, and Savage Attacker only trigger on melee attacks.

Factoring in additional attacks.

(Ignoring higher ability scores)

When fighters get Extra Attack (2), expected sharpshooter damage goes up to about 55.5 (Heavy) and 66 (Hand), but expected non-sharpshooter damage to 25.5 (Heavy) and 26 (Hand)

And Extra Attack (3) brings us up to sharpshooter 74 (Heavy) and 82.5 (Hand), with non-sharpshooter damage at 34 (Heavy) and 32.5 (Hand).

The Heavy Crossbow is better here.

Additionally, things like Action Surge, Haste, etc. also lean towards Heavy Crossbow

Factoring in higher Ability Scores (and damage increases)

Increasing your Dexterity to 18 or 20 (or beyond) are going to lean this towards the hand crossbow by the increase of the modifier (+1 modifier makes it +1 damage better, etc.). This is equally true of anything that's going to bump your damage. Magic ammunition, a vulnerable enemy, Elemental weapon, etc. are all going to have a similar effect, wherein more attacks is better.

Finding a special Hand/Heavy Crossbow

A +3 Heavy Crossbow of Awesomeness is almost always going to be worth using over most mundane Hand Crossbows.

Limited ability to take bonus actions

If something is limiting your ability to take your bonus action (slow, needing it for something else, etc), then that makes the Heavy Crossbow strictly better.

All together now

Almost always, for most characters, the Hand Crossbow is going to be better if you have Crossbow Master and Sharpshooter.

The exceptions to look out for are getting significantly more attacks (Fighter Action Surge, Fighter Extra Attack, Haste, etc.), extremely hard to hit enemies (I'm not going to do the math to figure out exactly what die roll you need, but high), and finding a magic Heavy Crossbow.

Additional Notes: BBeast, rightly, points out that if your opponent is outside of the Hand Crossbow's maximum range (120), but within the Heavy Crossbow's maximum range (400), then the Heavy Crossbow wins by default.

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    \$\begingroup\$ I actually started writing this to prove why the Heavy Crossbow was better... \$\endgroup\$ – goodguy5 Mar 7 at 3:04
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    \$\begingroup\$ I would also add the range of the weapons as an argument in favour of the Heavy Crossbow (30/120 vs 100/400 I believe). The Heavy Crossbow is better if you are attacking from beyond a Hand Crossbow's short range. \$\endgroup\$ – BBeast Mar 7 at 3:56
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    \$\begingroup\$ How about adding that a creature with a high Damage Reduction could be hurt more with the heavy crossbow? \$\endgroup\$ – Maxime Cuillerier Mar 7 at 6:30
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    \$\begingroup\$ @MaximeCuillerier because Damage Reduction - barring the very specific case of the heavy armour mastery feat - is not a mechanic that exists in 5e. Resistance is, but that just halves damage, so it makes a minimal difference how many attacks that damage is distributed between. \$\endgroup\$ – Carcer Mar 7 at 8:23
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    \$\begingroup\$ @Carcer thanks for the info, I'm unfamiliar with 5e. \$\endgroup\$ – Maxime Cuillerier Mar 7 at 9:13
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At a range of 121 to 400 feet

Your idea about why a hand crossbow does more damage is a pretty good reason as to why a hand crossbow with the sharpshooter feat does more damage than a Heavy Crossbow because of the bonus actions.... until you're too far away.

A Hand Crossbow has a range limit of 120 feet, while a heavy crossbow has a range limit of 400 feet.

The average enemy with a movement speed of 30 will take 5 turns to cross that distance of 280 feet if they're dashing, which means you could be shooting them for 5 turns before you could even hit them once with your hand crossbow.

After that, you can switch to your hand crossbow, but do not underestimate the ability to vastly out-range your enemy.

A range of 120 feet leaves you in range of a lot of dangerous spells and abilities. An Adult Red Dragon can fly 80 feet and then breathe 60 feet of fire, resulting in a toasty Sharpshooter who could have been at a safer distance.

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