In the first sentence, makes is actually in the indicative mood, not the subjunctive. That’s why makes has an s.
In the second sentence, go is in the subjunctive mood. That’s why it’s not goes.
Regarding the meaning of the sentences, both verbs describe a required situation that may or may not actually occur. The first sentence doesn't say that the bride's father is making a speech or will make a speech; it only says that he is required to. This is what calls for the subjunctive mood. When the wedding really happens, maybe the bride's father will do what etiquette requires of him, and maybe he won't.
However, grammatically, the subjunctive mood is mostly optional in English. People usually usually put the verb governed by require into the subjunctive mood, but not always. In other words, these sentences mean the same as your examples:
Etiquette requires that the bride's father make a speech.
The situation requires that he goes there.