Converting Analog Filter into Digital Filter, why Bilinear Transform?
$$\left . H_d(z) = H_c(s) \right |_{s = \frac{2}{T\,s}\frac{z-1}{z+1}}$$ describes a transfer function in the $z$ domain that you can easily translate into a difference equation and realize in software.
$$\left . H_d(\omega) = H_c(\Omega) \right |_{\Omega = \frac{\omega}{T\,s}}$$ describes an idealized frequency response that you would like $H_d$ to have when you are done realizing it physically in software.
They're different.  Note particularly the &quot;would like to have&quot; -- any translation from the continuous-time domain to the discrete-time domain is an approximation; part of your job is to make sure it's both practically realizable and good enough.
Note that there are other ways of approximating $H_c$ with some $H_d$ -- the bilinear transform is just one way.  It has a lot of currency because it's conceptually simple, and works pretty well.  It also has a lot of currency because it's easy to do with pencil, paper, and a slide rule -- today, there's numerical optimization techniques that can get you closer to some desired filtering goal, for less work (but -- I can never remember the search terms :( )