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Not so easy to awnser in short....
It is difficult to understand the relation between Vx & Vy (Vx = best angle and Vy = best rate of climb), without first understanding the effect of drag on an aircraft. If you look at the relation of total drag/power available relelative to velocity, you can determine different kinds of velocities such as Vx and Vy.
To achieve the highest angle of climb (highest nose position above horizon) for a propellor driven aircraft, you need the highest amount of power available of your engine. The highest power availability occurs at a speed which is lower than your minimal drag speed (Vmd), which would be your best glide speed or best rate of climb. At point Vx you have the most excess thrust availablewhich is the single factor enabling an aircraft to climb.
Note that Vx is a lower speed than Vy, your best rate of climb! Considering this, an aircraft reaches an height faster with the best rate of climb, while Vx has a higher nose attitude! Practicle use would be, to avoid an obsticle you could use Vx, and climb at shorter ground distances etc. Note that the use of flaps increase your Induced drag because the increase of lift, thus changing the total drag (reduces drag speeds) curve and changing the whole Vx/Vy speed spectrum... :)
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