At a theoretical level, D flat and C sharp are not the same note. At a practical level, depending on instrument, temperament and performance style, they might not be the same actual pitch. This could mean various things:
For a composer writing tonal music it is important to know the context of the notes you are using so that you can write appropriate harmonies, and to understand the structures you're working with so as to write music which fits with the style you're using. Thus we need to understand the differences between enharmonic equivalents because if you're composing in D major a D flat (flattened tonic) implies very, very different things to a C sharp (which is the leading note and a perfectly ordinary thing to see)
For a performer the use of sharps and flats gives us the same clues the composer was working with. As we develop in our knowledge of the theoretical ideas behind the music we're playing, we can then use those clues to help us play the music in a more understanding way. This is of especial and vital importance where the performer is expected to harmonise the written line or to improvise around it, such as in baroque music or the obvious modern example of jazz.
And they aren't even really the same pitch. Pianos compromise because it makes them feasible to build, transport and play. Violinists don't have to worry about this, as they can place their fingers anywhere they like; neither do singers. Most wind instruments allow a level of flexibility in the pitch for each fingered note and skilled players can exploit this.
Why would they do this? Because you can make the music sound better if you understand what the interval relationships are meant to be in a perfect world where nobody had to use compromised tuning systems for the sake of practicality, and can adjust their notes so that they get not just a proper D flat but exactly the right D flat for the current situation in the music. This is all based on frequency ratios, and turns into a much bigger discussion about the compromises inherent in temperaments and how you can almost never avoid them.