Is there a difference between the words "accounting" or "accountancy", when referring to the field or branch of study?
They are more or less interchangeable. Accounting has different meanings and in one sense refers to the process of keeping or maintaining financial records; while accountancy, according to OED, specifically refers to the duties(which consists of accounting, bookkeeping, auditing, etc.) or profession of an accountant.
When it comes to which is the child and which is the parent, accountancy seems to be the parent term in finance. Here is the definition from Wikipedia:
Accountancy is the process of communicating financial information about a business entity to users such as shareholders and managers.1 The communication is generally in the form of financial statements that show in money terms the economic resources under the control of management; the art lies in selecting the information that is relevant to the user and is reliable.[2] The principles of accountancy are applied to business entities in three divisions of practical art, named accounting, bookkeeping, and auditing.[3] The American Institute of Certified Public Accountants (AICPA) defines accountancy as "the art of recording, classifying, and summarizing in a significant manner and in terms of money, transactions and events which are, in part at least, of financial character, and interpreting the results thereof."[4]
That said, when referring to the field itself, I'll never use accountancy and will stick with accounting.
