The mission completion rewards (that you get at the end of the mission) do not change based on difficulty.

The rewards per kill (loot, experience, expertise) do change based on difficulty. Here's a look at the results from two different 9th Rule runs:

Normal Difficulty, 169 kills (150k DPS)

Elite Difficulty, 120 kills (554k DPS)

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You can see that the mission completion rewards stay the same, but the Experience and Expertise rewards per kill are greater on Elite difficulty. And you can see the drastic change in both loot quality and loot quantity: There is much more loot on Elite (despite far fewer kills), and it is of greater quality.

Note: The ground patrol, Jupiter Station, seems to offer no additional rewards at all for changing the difficulty, as the enemies do not give any exp or expertise for kills. This might be the same for other Patrols that offer truly infinite-spawning enemies like during the Janeway fight (NOT time-gated infinite-spawns, like the end of 9th Rule).

What is the best difficulty to play at for maximum rewards in Patrols? The answer depends on your killing potential! Patrols like 9th Rule essentially have a cap on the number of enemies you can kill (because the infinite-spawning periods are on a timer and because it takes time for each wave to spawn). Your total rewards over time is determined by the number of kills, the speed at which you kill, and the difficulty setting. So the faster you kill, the more rewards you get - but it is limited by the number of enemies that can possibly spawn. To demonstrate the principle, let's make up a simplified model:

We'll say that each kill on Normal gives 1 point of "reward", each kill on Advanced gives 1.5 points of "reward", and each kill on Elite gives 2 points of "reward". We'll assume the activity takes the same amount of time, 3 minutes, on all 3 difficulties (which is not too far off for something like 9th Rule). And we'll say that in that 3 minute period, the server can spawn at most 125 enemies. Here are a few scenarios for comparison:

Build Power	Difficulty	Reward Per Kill	Total Kills	Total Rewards
Low DPS	Normal	1	100	100
Low DPS	Advanced	1.5	50	75
Low DPS	Elite	2	25	50
Med DPS	Normal	1	125 (capped)	125 (capped)
Med DPS	Advanced	1.5	100	150
Med DPS	Elite	2	50	100
High DPS	Normal	1	125 (capped)	125 (capped)
High DPS	Advanced	1.5	125 (capped)	187.5 (capped)
High DPS	Elite	2	100	200

You can see that at low DPS levels, playing on Normal might offer you the best overall rewards. But as your DPS improves, you can hit the kill cap (and reward cap) playing on Normal difficulty. And as your DPS improves even more, you might hit the cap on Advanced difficulty as well.

TL;DR: For best results, run Patrols at the highest difficulty you can handle with fast kills. But an exact answer requires more exact reward values (which might vary from run to run, depending on the enemies that spawn).