I see a bookkeeper's responsibilities as getting the business transactions into the company's general ledger. This involves a tremendous amount of accuracy and persistence in first getting the information and then getting it entered. At smaller companies the bookkeeper is likely to process the payables (receiving suppliers' invoices, verifying them, and remitting the amounts), receivables (billing customers, processing receipts, sending statements), payroll, and other tasks. In larger companies, the bookkeeper's responsibilities are likely to be assigned to an accounts payable clerk, an accounts receivable clerk, and a payroll clerk. Generally, the bookkeeper (or accounting clerks) will not have a four-year accounting degree and will be paid considerably less than an accountant.
Accountants will review the information that the bookkeeper had entered into the general ledger, will prepare adjusting entries, will prepare the financial statements, and will analyze them. The accountant will likely supervise the bookkeeper (or accounting clerks), will be involved in the accounting system, and will review the financial statements with the management and owners of the company. The accountant will also be involved in budgeting of operations and capital improvements, cost accounting, reports to government agencies, and various analyses required by management. Generally, the accountant will have a four-year or a five-year college degree with a major in accounting.
Of course my remarks are a broad generalization. Responsibilities will vary by company and by individual.