They both mean the same thing. The second is a bit old-fashioned or poetic. To understand the second one properly, you have to understand the omitted words: “He cannot [do anything] but laugh.” You see this construction from time to time, but it sounds old-fashioned: “I cannot but cry.” “I cannot but think that . . .”

You need the word "help" in both sentences: 1. He can't help but laugh 2. He cannot help but laugh. The both work the same.

#1 not normally used in the third person in this way, though it might be appropriate in another context #2 sounds archaic [old-fashioned], we wouldn’t say that now