These two sentences have exactly the same meaning. If you really want to distinguish between them, you could say that the 'got' version implies slightly more influence or effort on your part. For example, a very small child could 'have' his hair cut. This simply means that he sat there while somebody cut his hair. You wouldn't say that the tiny child 'got' his hair cut, because he didn't really do anything. An adult, however, might say that he 'got' his hair cut, meaning that he went into town, found a barber's shop, told the barber the style he wanted, and then paid the barber after he'd had the haircut. Likewise, 'I got my room cleaned' might suggest that it was certain amount of effort for you to find someone to clean your room and to arrange for this to be done. That said, the difference is very subtle indeed, and these two forms are virtually interchangeable, It's also worth remembering that expressions with 'got' tend to be seen as more informal.

There's no difference in meaning. The one with "got" is more informal.