I have been asked this question many times and you will be surprised to find that most people (even folks from the Boxing industry) don't really understand it.
There is the technical definition between the two sports such as differences in the rules but that is a very shallow and limited understanding of the sport.
Fundamentally, amateur sports focus on finding the best athletes to represent the city, province, and finally the nation at various international tournaments for the glory of the country the athletes represent. This is a very important role which national Federations plays. The Federation can decide whatever it sees fit to devise local tournament rules and determine winners based on it's own standards since they are the sole power in the amateur sport they represent in that country. International community has no saying in the process of selecting top athletes to represent the country by each Federations but share same rules when competing in international tournaments such as ASIAD, OLYMPICS, SEA GAMES etc.
Professional sports however has a very different goal. It is all about commercial benefits to the organizer and the athletes participating in the event. It has no interest in finding the best boxer but rather focuses on what organizers can "sell" to the public and sponsors to watch. There are no international tournaments such as SEA Games, Asian Games, or the Olympics. In professional boxing, all matches are one single event and there is no medals or 2nd place award. There is simply a winner and a looser.
Professional Boxing is based on a global ranking system to determine if one boxer is "better" than another, and this has a big impact on the money that athletes can demand from the organizer for participating in an event. (Generally, if a fighter have had more fights in the past with many wins, they can demand more money because more people would likely know the boxer and thus it is easier for the organizer to market the event.) In this regard, the international community had to agree in using a common database that records such ranking. One of such platforms where you can find combat history of athletes is called BoxRec. It is a private website that has managed to invest in collecting most fight results around the world which is used as the first place to look when researching about a boxer.
Because of the global ranking, a boxing match cannot be called professional boxing simply because it applied professional rules. It also cannot be called as professional boxing by a national Federation or even by a government alone. For a boxing match to be accepted as "professional boxing", it must be recognized by international organizations as meeting certain standards, procedures, and paperwork. Otherwise, you can have any organizations claiming to have "local authority" to declare results of boxing matches that may not be fair and transparent. The professional boxing community has fought off countless attempts by various organizations claiming have "authority" to try and insert the results of matches that do not meet required standards.
Now who are these "international organizations" in professional boxing?
Most people think they are WBO, WBC, IBF, WBA. Many also wrongly think that above organizations are equivalent to IOC or FIFA but they are not. They are all private organizations that have been successful at marketing their brands around the world. There are over 65 of such companies but WBO, WBA, IBF, WBA are the biggest. Each of them have their own rules and criteria as to who can fight for their belt. They make money from the organizer (called "promoter" in boxing) who arranges events involving their brand. In any boxing event, they work with the organizer to witness and manage the fight involving their brand only.
Commissions are organizations that understand the requirements of being accepted as a "professional boxing" and act as an auditor to "supervise" a boxing event which are non-title fights. Almost all professional boxing commissions in the world are private organizations except for USA and the Philippines. Even in those countries, amateur boxing is separately managed from professional boxing since mixing amateur and professional sports often lead to corruption.
Commissions have commercial agreements with other commissions after ensuring that they are mutually satisfied with the quality and the integrity of the supervision (audit) of a boxing match which may involve boxers under their management. Each country can have multiple commissions. Korea has over 5 professional boxing commissions. Japan had 2 commissions at one time but now has one private commission. As long as a boxing match meets a certain criteria, the international community will accept it as being a "professional boxing match" and record the result.
In Vietnam, only VBO is accepted as having met international standard by other private commissions in Japan, China, Korea, Philippines and also Boxrec. This does not mean VBO is the licensing authority for a boxing event nor an exclusive organization for boxing in Vietnam. VBO works with any promoters who is licensed by the government to organize a boxing event to provide consultancy, audit, and boxing management to meet a certain standards that are recognized by boxing organizations which VBO has commercial agreements with. Only then, the boxing match is recognized as being a "Professional Boxing" match by the the international community.
It is important to note that if a boxing match is supervised by an organization or even government that the international community does not recognize as having a certain knowledge and standard, it will not be recorded as Professional Boxing no matter what the local organization claim it to be.
Recently, Vietnam Boxing Federation (VBF) has been trying to disrupt legally licensed boxing events claiming to be the sole authority to approve professional boxing in Vietnam. It is important to note that they have charged promoters in the past over USD 10,000 dollars for "sanctioning fee" and demanded various perks such as free rooms and flight tickets for their guests. They used amateur scoring system and ultimately had no recognition by international community involved in professional boxing. For these reasons and many more, all the professional boxing events they supervised in the past were not recognized by major commissions and boxing organizations such as BoxRec. In contrast, VBO provides its services for free to all promoters as a sponsorship.
Another reason why VBF is rejected by the professional boxing community is because some senior executives of VBF are also gym owners, managers of boxers, and also a promoter that competes with other organizers of professional boxing in the market. Clear conflict of interest is one thing but they haved used their authority to send various government departments to intimidate and try to disrupt boxing events by all other organizers.
So far, VBF has yet to supervise one single professional boxing event listed on BoxRec. In 2023, there were 2 events done by SSC. The owner of SSC is also the Vice President of HBF and President of VBF. The promoter of the event, Shadow Entertainment, is partly owned by the President of VBF.
Rather than just repeating an empty slogan that VBF is the sole authority, it is important for them to first act as a responsible organization by establishing a clear rules of professional boxing for Vietnam, and also establish working agreements with various professional organizations around the world so their supervision of a professional boxing match is accepted as credible and transparent.
In comparison, VBO does not have any boxers under contractual management and certianly don't charge any fees of any kind to boxers. It has sponsored over 13 events in 2023 with combination of free services to organizers, gloves, hand wraps, seminars, fighter purses, flight tickets, hotel, and meals. The result is that VBO has produced over 80% of new professional boxers in Vietnam including 100% of all female boxers. With over 2bil VND in sponsorship, it is unlikely VBO is out to collect money from boxers they are paying and supporting in the first place as claimed by VBF.
VBO does not have any form of agreements with WBA, WBO, IBF, WBC or any other such organizations. Those are commercial arrangements with the organizer/promoters.
"Sanctioning" is not a legal word. It means following a certain standards for a match to be recognized by the Sanctioning organization. For WBO title fights, WBO will be the Sanctioning organization for their brand and VBO does not get involved in that.
Function of a commission is to protect safety and welfare of boxers rather than showcasing their authority or titles guven by higher powers. It is important that any organization claiming to be the authority first demonstrate their capability before asking for the acceptance of their competence.