Both answer the same question — what do I do with an urge I should not act on — and they answer it in opposite directions, which is why it is worth knowing both.
Urge surfing does nothing. You observe, you describe, you wait, and the urge subsides on its own schedule. It asks for tolerance and it is passive by design.
Opposite action, which is chapter 10 of the book, does the reverse of what the urge demands. The urge says withdraw, so you make contact. It says attack, so you soften your tone. It asks for a behaviour, and it changes the emotion by changing what you do.
The choice between them is mostly about intensity and about you. Surfing is better when the urge is high and any action would be the wrong one — 2am, phone in hand. Opposite action is better at moderate intensity, when you have enough capacity to do something deliberate, and it is generally the faster of the two at actually shifting the feeling rather than outlasting it.
If you are working from the book, opposite action is what it teaches and it is worth learning properly first. Urge surfing is a reasonable addition for the situations where doing the opposite is not available and doing nothing is the entire task.
One waits the urge out. The other does the reverse of what it asks.