Speeding Ticket Application
It is a question of how much will the project expand beyond what is described here.  For instance, are you going to persist driver details and other data? if you are going to use a database, then you will probably want to use an Object-Relation-Mapping (ORM) framework, and they tend to expect Java Beans to work with. perhaps you will want to develop a web api that will send or receive such data in json format? json parsers/generators also prefer objects that follow Java Bean standard.

for very simple, small, one-man projects it is probably ok to skip the boilerplate code of getters/setters.  it is just that usually projects tend to grow and expand in time.

another point that can and should be addressed now is the calculator: it feels wrong to me to create instances just to do fine calculation. if we make the analogy to windows calculator, it is like openning a new window each time you want to do a new calculation.  the calculator is stateless: it takes arguments and produces a result. it does not carry any state between calculations.  So I would make the calculateFine() method static and have it accept the two arguments.

another issue is the calculation process itself.  all the attributes of a fine are closely related to each other and come in mutually exclusive sets: the result is a textual description of the enum, the amount and points are also linked to it.  

I would make these values properties of the enum:

public enum FineType
{ 
    NONE("The user was not going above the speed limit, no actions will be taken.", 0, 0), 
    WARNING("No fine issued, no points gained but a warning will be issued to the driver.", 0, 0), 
    FIFTY("A fine of £50.00 will be issued to the driver. 0 points gained.", 50.00, 0), 
    // and so on

    private String description;
    private int    fine;
    private int    points;

    FineType(String description, int fine, int points) {
        this.description = description;
        this.fine = fine;
        this.points = points;
    }

    public String getDescription() { return description; }
    public String getFime() { return fine; }
    public String getPoints() { return p[oints; }
}


now you can have the Fine take a type in the constructor and fill the properties by itself. this makes the calculator process clearer:

public static Fine calculateFine(int speedlimit, int carspeed) {

    int milesover = carspeed - speedlimit;

    if(milesover &lt; 1) { 
        return new Fine(FineType.NONE);
    }
    else if(milesover &lt; 5) {
        return new Fine(FineType.WARNING);
    }
    ...