I like to think of this in terms of what you can accomplish in the course of a mediation. Mediations generally have several things going on at once. They therefore usually have more than one type of outcome when they work well. Knowing the possible outcomes can help you make choices that will work for you.
People usually come to mediation with concrete issues in mind. This may include money, division of property, child custody, or who has what responsibility in a business relationship. You may have ideas about what looks fair to you in terms of who pays whom what, or who gets which part of the assets, etc. Helping parties to a dispute find specific solutions to such issues is where problem solving skills come in handy. They include being able to help you pick out the key issues, brain storming, setting priorities, etc. These skills allow the mediator to help you focus on what is important to you, and to come up with solutions that you can live with.
Rarely do people bring such issues into a mediation in a purely objective, unemotional manner. Things are important to us as human beings. Emotions can help us figure out what is important to us. They can also prevent us from seeing things from a different perspective. If you feel angry, worried, or frustrated, this can become a barrier to resolving your problems. This is something both the problem solving and the Transformative mediation models address. The mediator needs to have skills in communication. This will include listening so the parties feel they are being heard. If people think no one is listening to them they tend not to listen in return, but to repeat their own viewpoint. When we are angry, we tend to add judgmental words and phrases to what we say. The other person focuses on those judgements, not on what it is we really want. The mediator must be skilled at rephrasing your words so that the essential meaning is not lost, from your perspective, and the judgmental wording is removed, so the other person can hear what you are trying to say. Again, these skills help you to focus on what is important to you, and away from anything that distracts from reaching agreement.
Often, disagreements escalate because of difficulties in the relationship between the parties. The things that most often get in the way are lack of clarity, and a feeling that the other person doesn’t see how it is for you. This is the domain of the Transformative model. When we aren’t clear about our own needs, values, choices etc. It is more likely that we will be defensive and unwilling to compromise. Likewise, if we feel like the other party just "doesn’t get how it is" from our perspective, we will be more likely to insist on pushing our own agenda, rather than really listening to each other. The mediator’s job is to help us clarify our own thinking, and to get how things look from the other person’s perspective, even as we continue to disagree with it. This work is about helping the working relationship to improve for the future. This can be valuable to you even if you don’t anticipate a continuing relationship with the other party. Anything you learn in terms of how to be clear for yourself, and how to understand where the other person is coming from will help you avoid seemingly unresolvable disputes in the future.
All of these are potential positive outcomes of a mediation. You needn’t worry so much what model the mediator tends to follow, if they are skillful in all of these areas. Most of us tend to start from one or the other model because of our background, training, personal inclination, etc. The key, in my opinion, is for the mediator to be aware of all the pitfalls and promises of your conflict, and to facilitate your dialogue with the other party to the dispute in each of these skill areas. When you interview a mediator, ask how they would describe their approach. What are their strengths? Where were they trained, and under which approach? How do they present themselves in the interview? What kinds of questions do they ask. How do they respond to what you say about the conflict and the other party? Mediation is an extension and condensation of real life. The way they act when you interview them is one indication of how they will conduct themselves in the mediation.