Voltage across diode, Shockley equation
There are various diode models of varying fidelty to reality and I think you are mixing two of them.

The model you describe as


  I know that every diode has a small potential difference and only when the applied voltage is greater than that voltage across the diode, electrons are able to cross the barrier. 


differs from the model given by the Shockley equation. So don't be surprised if you get incompatible results if you mix them.

The Shockley equation is more accurate because in reality a diode is not either on or off but it conducts already little bit below the threshold voltage and also above it current increases more and more. 

See following i-vs-v graphs of diode models that get more and more realistic. Note that although they may give you different results none of them is totally "wrong" or "right" just more or less accurate.



Back to your question:


  is the applied voltage same as the voltage across the diode?


Applied voltage is always the voltage across the device you apply it to. So this question doesn't make sense.

I guess you mean "Is the voltage across the diode the same as \$V_D\$?"


For the simple models (top row) if the diode is on the voltage across the diode is always \$V_D\$ (constant; independent of current). There is nothing to calculate.  
Only for the more accurate models (bottom row) you can calculate the voltage for a given current and it is not the same as \$V_D\$ (depending on the current).