"And we have known and believed the love that God hath to us. God is love; and he that dwelleth in love dwelleth in God, and God in him." (1 John 4:16)
Aren't they basically the same thing?
"To know" stands in opposition, as in "not knowing."
Many people simply do not know that God loves them. They think that God is angry with them, or that His love for us is dependent on what we do, as opposed to what He is doing, or more importantly what He has already done through His Son.
"To believe" means that you stay confident in God's love for you, even when things are not going your way--so it seems.
When it seems that God is not on your side, He is very much on your side. When it seems that everything is falling apart all around you, you rest in the faith that He is holding everything together for you.
"And he is before all things, and by him all things consist [are held together]." (Colossians 1:17)