When you use a const string, the compiler embeds the string’s value at compile-time.
Therefore, if you use a const value in a different assembly, then update the original assembly and change the value, the other assembly won’t see the change until you re-compile it.
A static readonly string is a normal field that gets looked up at runtime. Therefore, if the field’s value is changed in a different assembly, the changes will be seen as soon as the assembly is loaded, without recompiling.
This also means that a static readonly string can use non-constant members, such as Environment.UserName or DateTime.Now.ToString(). A const string can only be initialized using other constants or literals.
Also, a static readonly string can be set in a static constructor; a const string can only be initialized inline.
Note that a static string can be modified; you should use static readonly instead.