In an undated sermon on Ps 147:1, Edwards says there are three kinds of praise: praise with our behavior, praise with our lips, and praise with the heart. He cites Scripture in support of each.
But then he says that the first two only have worth if they include the last one. He explains why by describing exactly what praise in the heart is.
Praise in the heart is an inward expression of love, joy, and admiration toward God under a sense of his excellency and grace. . . .
There may be admiration where there is no praise. The devils doubtless wonder at many things which God does. 'Tis probably they were surprised and amazed at the incarnation, death, and sufferings of the Son of God. Such a manner of expression of grace was what they did not in the least expect when man fell . . . but yet they don't praise but blaspheme. There may likewise be joy because of God's goodness when there is no praise because their joy does not arise from, nor is accompanied with, love. Wicked men rejoice in many things that are fruits of God's goodness to them, but they don't praise him because the foundation of joy is not anything that they see in God but only the good they receive in themselves.
Therefore then only is God praised when the heart is lifted up to God with love, joy, and wonder for what it sees in him and receives from him with a desire of expressing it to him.
--Jonathan Edwards, "It's a Very Decent and Comely Thing That Praise Should Be Given to God," in The Glory and Honor of God: Vol. 2 of the Previously Unpublished Sermons of Jonathan Edwards, 124