Diffusion current
When a p-n junction is formed, a diffusion phenomena causes electrons from the n-doped region to diffuse to the p-doped region. At the same time (even if it's an abstraction) holes diffuse from the p-type region to the n-type one. The atoms that lose a carrier (electron or hole) become ions, which means that instead of being neutral, they have a positive or negative net charge. This happens because the ideal equilibrium would have the same concentration of mobile carriers equal all over the region.
Ohmic current
However, this diffusion causes the growth of a region, populated by ions, called depletion region, because all atoms have lost their carrier. These ions, as we said, are electrically charged, and cause an electric field directed from the n-region to the p-region, pushing carriers in the opposite way than diffusion. Therefore an equilibrium is reached in which the current (movement of carriers) caused by diffusion is perfectly balanced by the current caused by the electric field (ohmic current).
Effect of biasing
Applying a potential to the junction causes a perturbation on this equilibrium, making one of the currents dominant on the other. Reverse biasing the junction causes the ohmic current to prevail, while forward biasing increases the diffusion current.
Now, the diffusion current is a much stronger phenomena, from which derives the exponential growth of the forward bias current with the bias voltage. Ohmic current, on the other side, is much weaker, and saturates quite soon (neglecting avalanche effect) because the width of the depletion region (which determines the resistivity) is proportional to the reverse bias voltage.
If you attempt to apply a forward bias voltage larger than the diode's "natural" forward voltage, the diode will draw more current in an attempt to bring the applied voltage down to its "natural" voltage. If the power supply is sufficiently "stiff" (maintains its output voltage regardless of current), the diode will release its magic smoke, and become an open circuit.
Best Answer
I think, there is a small confusion on your side regarding the meaning of "barrier voltage".
Caused by a diffusion process across the pn-junction there is a "barrier" consisting of a potential difference (called voltage). This "built-in voltage" causes the diffusion to stop at a certain equlibrium. Hence, "barrier voltage" and "built-in voltage" are two different expressions for the same quantity.
For a current to exist through the device there must be an external voltage which is able to reduce this barrier. Thus, this voltage further supports the diffusion process and allows a exponentially rising current. This effect is visible already at an external forward voltage of app. 0.1 Volt).