Li-zchus - 
R' Moshe Yehuda Hanus
R' Shmuel Stein 
R' Chaim Schreck 
R' Eytan Feldman
R' Avromi Sommers
For much success in all they do together with their families

A person who was ruled by a Beis Din as subject to capital punishment is already considered a dead man - "גברא קטילא". As Rashi writes [Sanhedrin 71b]: 

"משעת גמר דין הוי כגברא קטילא". [ועי' צ"פ הפטרת ויחי].

The Meshech Chochma [Yisro] teaches that this means that one who harms a גברא קטילא is exempt from punishment because the person lost his right to exist:

"כבר בארתי בחידושי שרציחה לא הותרה מכללה בב"ד כמו חובל בחבירו משום כיון דנגמר דינו לאו אדם הוא ואין עליו שם אדם והחובל בו וההורגו פטור וזה רק גמר דין שאין לו משפט אנושית."

This is in contrast to a person who was given the death penalty by a king who is not considered dead and a גברא קטילא until he is actually executed. This is how the Rogochover explains the Rambam [הלכות אבל א ט]:

"כל הרוגי מלכות אע"פ שנהרגו בדין המלך והתורה נתנה לו רשות להרגן הרי אלו מתאבלין עליהן ..... אבל כל הרוגי בית דין אין מתאבלין עליהן." 

"We observe mourning rites for all of those executed by the government, even when they were executed by the government's laws and the Torah granted it license to execute them....We do not, by contrast, observe mourning rites for those executed by the court."

A person who was killed by the courts was already dead before they killed him and thus there is no one for whom to mourn. The very crime he committed made him a גברא קטילא and the judgment just exposed this reality. [This seems to  be different than the Rashi we quoted who says that he only becomes a גברא קטילא when the גמר דין is completed - דעת עצמי].  In contrast, a person killed by a king had the status of a viable person and thus mourning is mandated. 

The rights of a king to execute are limited. He may only kill with a sword [Hichos Sanhedrin 14-2] whereas a Beis Din may kill in any way they can if they are unable to carry out the form of execution that this person has coming to him [Sanhedrin 14-8]. 

A king only executes based on his own will [Melachim 3-8]. Therefore a person who is about to be executed by a king may be assessed for his value [ערכין] for the sake of a vow because the king might change his mind [even if he couldn't change his mind the ערך should be valid because he is still a living being - דעת עצמי] while someone who is to be executed by the Beis Din may not be assessed because it is not the court that is executing him based on their will but rather it is the Torah who commands them do so [רמב"ם פיהמ"ש ערכין פ"א].       

[עפ"י משנת רבינו יוסף הרוגוצ'ובר]