In some situations, Intimidate can be successful on a much lower roll than Diplomacy. RAW, the DC for an Intimidate check is 10 + the target’s Hit Dice + the target’s Wisdom modifier, and success automaticallyalways makes the target Friendly toward you for a short amount of time regardless of their original attitude.
By contrast, Diplomacy with an Unfriendly creature has a DC of 20 + creature’s Cha Modifier, and a success only improves their attitude to Indifferent; you'd need to succeed by 5 more to improve it to Friendly. Diplomacy with a Hostile creature often isn't possible, but it has a DC of 25 + creature’s Cha Modifier, and usually only improves their attitude to Indifferent at best:
Diplomacy
Succeed – If you succeed, the character’s attitude toward you is improved by one step. For every 5 by which your check result exceeds the DC, the character’s attitude toward you increases by one additional step. A creature’s attitude cannot be shifted more than two steps up in this way, although the GM can override this rule in some situations.
Of course, this can work in the other direction as well. Because the target's Hit Dice affect Intimidate DCs but not Diplomacy DCs, Diplomacy may have the lower DC when interacting with high-level targets (again, RAW - it's not uncommon for published adventures and DMs to alter these rules).