The APY calculates the annualized rate for a savings product and adjusts for compounding.  It assumes a balance of $1,000 for a 12-month term. It also assumes no withdrawals are made during the 12 month term. Truth-In-Savings (TIS) regulations require this disclosure to help people compare savings products, apples to apples, across multiple financial institutions.
The APYE calculation is also required by TIS, and can be reported to members via their statement. It is the annualized rate that takes into account both the dollar amount of dividends paid for that period and the average balance of the account for that period.  In other words, instead of assuming a generic balance of $1,000 and a 12-month period, it looks at the account's actual balance and dividends paid.