Yes, the sentences certainly are different in meaning. It's the difference between forcing someone to do something ("to make" them do it) and requesting that they do it ("to have" them do it). The verb "to make" is very strong—to "have" someone do something is more like a request. Say I'm the boss at an office and I'm giving a task to an employee. There's a big difference between saying:
"Could I have you do this for me?"
"Could I make you do this for me?"
With the latter pair of your example sentences, "I had him cry" is a very odd thing to say. It certainly doesn't mean "I made him cry"—it means something more like "I directed him to cry". Again, the difference between forcing someone to do something and requesting that they do it.