The compiler will throw an error if you try to assign a value to a constant, thus possibly preventing you from accidentally changing it.
Also, usually there is a small performance benefit to using constants vs. variables. This has to do with the way they are compiled to the MSIL, per this MSDN magazine Q&A:
Now, wherever myInt is referenced in the code, instead of having to do a “ldloc.0” to get the value from the variable, the MSIL just loads the constant value which is hardcoded into the MSIL. As such, there’s usually a small performance and memory advantage to using constants. However, in order to use them you must have the value of the variable at compile time, and any references to this constant at compile time, even if they’re in a different assembly, will have this substitution made.
Constants are certainly a useful tool if you know the value at compile time. If you don’t, but want to ensure that your variable is set only once, you can use the readonly keyword in C# (which maps to initonly in MSIL) to indicate that the value of the variable can only be set in the constructor; after that, it’s an error to change it. This is often used when a field helps to determine the identity of a class, and is often set equal to a constructor parameter.