Diode built-in voltage and barrier voltage (forward bias voltage) are they almost equal? how do they relate to each other?
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).