I was recently told that using Enum:

public enum TaskEndState { Error, Completed, Running }

may have compatibility/serialization issues, and thus sometimes it's better to use const string:

public const string TASK_END_STATE = "END_STATE";
public const string TASK_END_STATE_ERROR = "TASK_END_STATE_ERROR";
public const string TASK_END_STATE_COMPLETE = "TASK_END_STATE_COMPLETE";
public const string TASK_END_STATE_RUNNING = "TASK_END_STATE_RUNNING";

Can you find practical use case where it may happen, is there any guidelines where Enum's should be avoided?

Edit: My production environment has multiple WFC services (different versions of the same product). A later version may/or may not include some new properties as Task end state (this is just an example). If we try to deserialize a new Enum value in an older version of a specific service, it may not work.