God, as our Father, wants and provides the same for us and even more so because He is God. God made us in his image and likeness. We can, through the example of Jesus and applying the Word of God to our lives, grow up to be like him. Because of natural parents, we might say of someone, "She has her father's eyes and his smile." How much greater the compliment if we can say, "She has her Father's eyes and smile, his compassion and love of others." Or, "He has learned patience and self-control."
As children of God we hold a certain place in society. After all, our Father is the Supreme Godhead, Creator and Ruler of the Universe. If we know who He is, then we should know who we are. We can brag on our Father and bring people to meet him. People might look to us to be leaders and to grant favors which we would be able to offer through exercise of the fruits of the Spirit (Galatians 5:22).
As children of the Father, co-heirs with Christ, we inherit all that he has. Jesus said, "All authority in heaven and on earth has been given to me" (Matthew 28:18) and he passed it on to his disciples. He said we could drive out demons, speak in new tongues, and heal the sick. And we will inherit the Kingdom. At the end He will say, "Well done, good and faithful servant. Come, enter into my joy (cf. Matthew 25:23).
That's what it means to be an heir.