Typically, NPCs will have some degree of opinion about you. They may be friends, enemies, random bystanders, sycophants or someone you accidentally insulted.
It depends on how much time you have to spend. Diplomacy typcoslyltypically represents a reasonablyreasonable amount of back and forth negotiation that could span several minutes whilst Intimidation can be as quick as a single threat and a show of steel.
When using Diplomacy, you also typically take their opinion into account for the DC. They may be friends, enemies, random bystanders, sycophants or someone you accidentally insulted. After all, it's much easier to convince your old drinking buddy to let you into a restricted area than it would be to convince the xenophobe who thinks adventurers are little better than hobos.
Conversely, Intimidation doesn't care how much the person likes you, it depends on whether they believe you can make good on your threat. Of course, if you regularly intimidate your friends, they may not be your friends for much longer.
This we come to our final difference. Consequences. Intimidation is typically quicker and easier for your average burly Warrior to get what he wants, but it can leave a mess. Diplomacy takes longer and make cost concessions, but is less likely to bite you in the long run.