Another way that dissents matter is when they have great persuasive influence in other, related cases.
A bit of quick background: the “holding” of a court case is the rule of law it creates. That is the part with actual, binding legal force–in a supreme court case, the part that lower courts are compelled to obey as precedent.
“Dicta” is the term for other stuff in the opinion that is not necessary for the holding–for example, suggestions as to how the rule might be applied, or a discussion of how the case might have come out if the facts were different (as a way of limiting its effect), or discussion of the policy behind the holding.
The point is that, while dicta is not precedent–it’s not something lower courts are obliged to obey, it has a lot of influence–it may suggest how the supreme court will rule, or be persuasive.
But dissents can do that too. They can be better, or more persuasive than dicta in the opinion. And in that regard, they matter a lot.
So, for example: let’s imagine a hypothetical case about how to apply a law banning “motor vehicles” in a park.
In the case, there is a 5-4 decision. Justice Douglas, writing for the majority, holds that a lawnmower is a “motor vehicle” because it has a working motor.
The opinion (still Douglas, J, writing for the majority), goes on to suggest in dicta that it’s not that important that the motor be working–that the reasoning would apply to a car with a non-working motor. This is dicta because it’s not necessary to decide the case—it’s a hypothetical about how the reasoning would apply to a different case to which the holding wouldn’t apply.
In his dissent, Justice Holmes disagrees with the reasoning of the holding–that in his opinion, a “motor vehicle” is something that people ride on. He also disagrees with the dicta, arguing that it’s crazy to call something a “motor vehicle” when the motor is a useless lump of metal–that in his opinion, the motor must be working.
Now, a District Judge is deciding a case about an antique tank put up as a monument–whether it is banned as a “motor vehicle”. It has a motor, but the motor doesn’t work. If the holding did apply, he’d have to follow it–to listen to Justice Douglas, and ignore Justice Holmes.
However, in this case, holding doesn’t apply–the tank’s motor doesn’t work. So there is no “rule” he has to obey. He has to decide something the Supreme Court has talked about, but hasn’t ruled on–whether something with a non-working motor is a motor vehicle.
In our hypothetical, the district judge may agree with Justice Holmes’ reasoning–that it’s crazy to call something a “motor vehicle” when it doesn’t have a working motor.
In cases like that, dissents can be very important. The dissent does not bind the lower court–it will never have to obey it–but it might be very persuasive, and have a lot of effect in deciding cases similar to the Supreme Court case.