So, this is about control.
Largely, the two systems are alike. They both grant permission to an organization to operate a railway service. The difference is in how the relationship is defined.
Franchising typically is about giving license to a franchisee allowing them to use intellectual property, know-how, products and so on in exchange for a fee and adhering to a set of obligations. The franchisee then is free to setup a business selling branded services and products.
A concession contract is different. It's an (exclusive) right which is granted to a concession holder giving permit to use an asset required to operate a service. Between private parties, a typical example would be a concession stand in a sports stadium (permit is giving for you to operate a drink, food, souvenir stand on the premises). Usually, the concession holder will pay a fee to the concessionaire.
A public service concession is a subcategory, and, depending on local legislation, works in a different way. The management contract doesn't provide leeway for the concession holder as to how to operate the service. The management contract specifies almost everything: fares, timetables, performance indicators, awards/penalties and so on. The operator collects revenue on behalf of the public authority.
So, why sign up for a public service concession? Because the concession holder gets a predictable source of revenue through public funding.
The theoretical benefit to the public is that the mode of operations resides with a public body which holds responsibility, and therefor can be held publicly accountable in a direct way as to how railroad systems are operated. In practice, disparate political interests can (and will) influence and even cause friction in how these concessions are governed.
That doesn't mean franchising is necessarily the better option. In this model, public control over the entire operational aspect of railway servicing is largely relinquished to private actors, any requirements set towards franchisees only represents a minimum bar for quality of service which they need to attain.
Shifting dynamics within the private market of franchisees (actors entering/leaving/getting acquired/...) may also impact the overall reliability of services. In a concession model, there's an (perceived) assurance that the same actors will continually operate a service for a predefined period of time according to set requirements.
As to which model is "better". I think exploiting national railway services is a complex problem domain which requires specific business expertise. Beyond the occasional reading online, I can confidently say that I'm absolutely not in a position to attribute absolute value to one over the other. Especially not at a point where information for this concrete case publicly available is limited to a white paper.