Clearly, closed primaries must be compared and contrasted with some other form of selecting people to run for office. I will assume that you mean for us to compare closed primaries with open primaries.
The major similarity between these two kinds of primaries is that they are both elections as opposed to caucuses. This means that a relatively large number of people will be able to participate in them. It also means that the candidates will be selected by the general public and not by the inner circles of the parties.
What is different about these two is who is allowed to vote in them. In a closed primary, only the members of a given party can vote. In an open primary, anyone can vote. That is, a person who is usually a Democrat can vote in the Republican primary or vice versa. This can be a bad thing for a political party because it allows their enemies to help select who will run as the representative of that party. For example, Democrats in the 2012 presidential election could have crossed over and voted as Republicans to try to elect the worst possible opponent for President Obama.
Thus, these two kinds of primaries are both more open than caucuses, but they differ in terms of who is allowed to vote in them.