Urs Hölzle, the man in charge of Google’s high-profile cloud-computing business, likes to tell colleagues that five years from now, they’ll “laugh about what the cloud was like” in 2015 because it will have evolved so drastically.

How much Mr. Hölzle shapes that evolution will determine whether Google will have a big business in selling computing infrastructure to other companies, something that it’s building for monetary and other strategic reasons.

But previously undisclosed financials as well as interviews with people who know the business show just how much Google lags behind the cloud-services leader, Amazon Web Services, and why its best hope seems to be a podium finish rather than outright winner.