What bond yield is measuring
Current yield is the simplest measure: annual coupon income divided by the bond's current market price. It is useful for a quick income snapshot, but it ignores how far the price is from face value and therefore ignores the gain or loss that can occur if the bond is held to maturity.
Yield to maturity is broader. It asks what annualized discount rate makes the bond's remaining coupon payments plus its redemption value equal to today's price. That is why two bonds with the same coupon can have different yields when they trade at different prices.