Yes in Some Dice Pool Systems
Modifying difficulty works in various ways within and between various roleplaying systems.
Indeed, not all systems are the same, and what is a distinction without a difference for a single-die engine that uses a d100 or d20 is a huge difference for a dice-pool-and-TN system like the Storyteller system (World of Darkness) or Exalted. I will show how and why it makes a difference in those systems.
Please note that this answer is written in reply to the system-agnostic variant of the question, as seen circa 2019-01-25, not in reply to system-specific versions.
How Storyteller Dice Rolls Work (Used as an Example)
The system uses a number of d10 for all rolls. The number of dice is determined by the character's sum of attribute and skill (usually the total ranges from 1 to 10), and some roll modifiers (e.g. using supernatural effects to increase Dexterity and/or aiming for extra turns before taking a shot can give you a few extra dice; conversely, being wounded penalises your roll, meaning you get fewer dice). The difficulty (target number) is dictated by the rules or the GM. For example, hitting someone with a punch is difficulty 6, while hitting someone with a punch in the dark is difficulty 8.
When you make a roll, you see how many dice come up equal target number or higher; this is the number of successes you scored. Each 1 rolled also subtracts one from your number of successes. Enemy actions, such as dodging, can further subtract from your number of successes. (There are potential other complications that I won't go into for the purposes of this example.) But so long as at least one success remains, you've . . . succeeded to some degree at your action. If you score more than one success, you succeeded better at your action - e.g. that may grant you a damage bonus on a to hit roll, or mean you not just finished making your film but also achieved more fame and earned more money from it than the expected minimum.
Why in Such Systems, the Difference Matters
Mathematically, on average, one expects a number of successes approximately equal to dice×(10-target number)/10. (Don't use this formula for precise calculations, especially for difficulty 9 or 10.)
This means that when the target number is low, adding a roll modifier (such as an attribute increase, or scoped aiming) will significantly increase the expected number of successes (almost by one success per die). Conversely, when the target number is high, like 9, adding extra dice has very little effect on the final result.
A similar relation exists on the other side: when the dice pool is low (such as due to having a -5 roll penalty from wounds), changes to difficulty have relatively little effect on the outcome, and you won't be scoring many successes even if lucky; when your dice pool is huge (e.g. 15 for a highly optimised sniper character taking a well-prepared shot), a single level of difficulty (target number) difference can provide more than one success worth of difference in expectation.
There is a fair amount of criticism against such a dice engine, which resulted in some subsequent systems by the same group of developers to have fixed target numbers (e.g. Chronicles of Darkness' Storytelling system has the target number fixed at 8 for all rolls, only using roll modifiers for everything), and a split in the community over which is more enjoyable in play.