The Australian Concise Oxford Dictionary defines the term "inventory" as;
"1. a complete list of goods …"; "2. the goods listed in this"; "3. the total of a firm's commercial assets".
Asset Inventory is a list of all the commercial assets that a firm (an organisation) owns or possibly has control over (in the case of local government).
The asset register is where most of these assets (the primary ones) are listed and where they may even be "componentised" to meet statutory reporting and provide a breakdown consistent with financial reporting requirements.
The asset register may not (need to) list the total asset inventory but also may break down the assets into components for asset management and financial accounting purposes. The inventory list would normally contain the primary (high level) assets which would account for the components (pieces).