You've got the basic idea of using enum to separate what's matching, but there are a couple of mistakes here:

• Json schema arrays don't have properties, they have items.
• Within those properties you're defining name as an attribute that then holds other json schema objects.
• In the second branch of your schema you defined otherProperty3 but in your sample that property is called anotherProperty3

Try this slightly modified version:

{
"$schema": "http://json-schema.org/draft-04/schema#",
"type": "object",
"required": ["items"],
"properties": {
    "items": {
        "type": "array",
        "minItems": 1,
        "additionalProperties": false,
        "items": {
            "anyOf": [
                {
                    "type": "object",
                    "required": ["name", "otherProperty"],
                    "additionalProperties": false,
                    "properties": {
                        "otherProperty": { "type": "string" },
                        "name": { "enum": [ "foo" ] }
                    }
                },{
                    "type": "object",
                    "required": ["name", "otherProperty2", "anotherProperty3" ],
                    "additionalProperties": false,
                    "properties": {
                        "otherProperty2": { "type": "string" },
                        "anotherProperty3": { "type": "string" },
                        "name": { "enum": [ "foo2" ] }
                    }
                }
            ]
        }
    }
}
}