What is the difference between having a Constitution and being a Constitutional government?

To be a constitutional government you would need a constitution.

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The US has a famous constitution, even if its value and authority are eroding over time. It is a single document written for the express purpose of establishing a government based on the premise that governmental authority is derived from the consent of the governed, and ideally all other laws in the US are in line with constitutional principles. The United Kingdom is a constitutional monarchy, and therefore a constitutional government. The monarch is head of state and has a number of functions. The monarch has a little more power and authority than is widely known, but wisely monarchs seldom use them. The system in the UK is different from the US system in that there is no single document that is considered 'the constitution'. But the UK holds to a system of laws (some of them truly ancient, and the basis for much law in western cultures) and processes that guarantee the orderly exercise of government under a law that protects all.