Is there any practical or mathematical difference between player rolling vs. DM rolling?

Question

There are two ways that spells are rolled for in D&D 5e, depending on the spell: attack rolls rolled by the players and saving throws rolled by the DM.

But as far as I can see, saving throws could be made as attack rolls instead. I know D&D 4e did it this way, with players making attacks against AC/Fortitude/Reflex/Will Defenses instead of the DM making Fort/Ref/Will saving throws and players making attack rolls against AC.

I don't really see why the designers of D&D 5e would choose to have two different ways to roll something that, as far as I can tell, could be simplified into one mechanic (and I have asked about their stated reasons in a separate question), so I wondered if I'm missing something important that distinguishes the two methods.

I guess it boils down to: Is there any difference in probabilities or practical concerns between one or the other method, or are they interchangeable?

This is related to Why do attacks function differently to spells? and Why did saving throws return in D&D 5E? This question is focused on if there is any practical difference, regardless of why they actually made that design decision.

Answer 1

Yes.

There are a number of game mechanics in D&D 5e that interact with Attack Rolls/Saving Throws to create significant differences between the two. The most notable are Advantage/Disadvantage and Critical Hits.

Advantage/Disadvantage

Having advantage or disadvantage on any roll significantly changes the probability of success. While this mechanic can influence many situations in the game, there are two situations in particular where advantage/disadvantage creates practical differences between spells that call for an attack roll and those that allow a saving throw: when making ranged attacks in close combat, and when using inspiration

Ranged Attacks in Close Combat

From PHB p. 195

When you make a ranged attack with a weapon, a spell, or some other means, you have disadvantage on the attack roll if you are within 5 feet of a hostile creature.

This means that spells which require an attack roll will sometimes suffer from disadvantage in situations where those that require a saving throw do not.

Inspiration

From PHB p. 125

If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.

This means that spells which require an attack roll can benefit from the use of inspiration, while spells that require a saving throw cannot (you cannot use inspiration to give your target disadvantage on a saving throw).

Critical Hits

From PHB p. 194

If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target's AC. In addition, the attack is a critical hit ... If the d20 roll for an attack is a 1, the attack misses regardless of any modifiers or the target's AC.

This means that spells that require an attack roll have a chance to hit (or miss) regardless of modifiers, and a chance to deal additional damage. Spells that require a saving throw cannot score a critical hit.

The previous points are intended to show that attack rolls and saving throws are not simply interchangeable, due to the ways that they interact with the larger system. In particular, the rule regarding ranged attacks in close combat makes spells that require a saving throw a bit stronger at short range; replacing the saving throw with an attack roll would weaken them. Recall that D&D 4e, which uses only attack rolls, has to have additional rules for 'close' attacks in order to allow short-range spells. With saving throws in 5e, such rules are no longer necessary.

Answer 2

The difference between these two systems is who is given apparent control over the outcome of the random roll, and nothing more. This has more to do with psychology and what 'feels right' to your players than it has to do with any real mechanical difference.

If you were to, say, target a player with a spell that causes instant death and you make a random roll to determine the success, and succeed, a player would feel like they 'couldn't do anything' to avoid it. They died, and no event under their control could prevent it.

In the case of a defense roll, a player gets to throw the die, and that comes with a greater feeling of control, even though it isn't really different from the other system in terms of probability.

Where there may be a practical difference, though, would be in GM fudge factor where some GMs will alter the results of a random roll to achieve some end. Perhaps a player has cast their death spell on your big bad and they've surely killed him with a nat 20 and your story comes to an abrupt, unsatisfying end. Had you rolled to save behind the GM screen and gotten a 1, you could happily declare your big bad made the save and carry on as normal. Defense rolls give you the option, as a game master, to spare your NPCs from story destroying results, if you wish.

In short, any roll the GM makes is under their complete control. The GM may choose to use the random result or not. Any roll a player makes is under their apparent control, even if that is not actually so because the die roll they are making is random.

Mechanically, the difference is who gets a flat added to their stats vs. who rolls the die. In 3rd Edition the DC of a save is:

10 + Spell Level + Ability Modifier

The Saving throw itself is calculated as:

d20 + Class Modifiers + Ability Modifier

The d20 is opposed by the +10. To convert between attacker and defender making the roll, you swap these two values in the calculations.