Apreciar y agradecer
Apreciar is a false cognate of appreciate. Apreciar means to hold somebody or something in high esteem, think highly of or value somebody or something. To a much lesser degree does it mean to appreciate. Examples:
Aprecio a la familia Rodríguez. Son excelentes vecinos y amigos.
I think highly of the Rodriguez family. They are excellent neighbors and friends.
Aprecio la buena música, la literatura y un buen vino.
I value good music, literature and a good wine (appreciate could also be used here, but not with a sense of gratitude but taste).
Le tenemos mucho aprecio al párroco de nuestra iglesia. Es un hombre dedicado a Dios y a la humanidad.
We hold the pastor of our church in high esteem. He's a man dedicated to God and humanity.
Now for agradecer:
Agradezco toda la ayuda que me has prestado.
I appreciate all the help you have given me.
Juan y yo estamos sumamente agradecidos por todas las tarjetas de pésame que recibimos el mes pasado.
John and I are really appreciative for all the sympathy cards we received last month.