Private cloud operates like a public cloud only, but the resources are operated exclusively for a single organization.
The message to be conveyed is:
Private cloud operates just like a public cloud except that the resources are operated exclusively for a single organization.
The usage of "only" in first sentence seems wrong to me. In my opinion, the word "only" makes it sound more like:
Private cloud operates like a public cloud only and not like any other cloud.
Am I correct about the interpretation of the sentence or is it grammatically correct? If it's wrong, what would be the correct one-word-replacement of "only"?