Let's begin with some definitions. Sufism is a movement within Islam that emphasizes mysticism. Practices such as chanitng, meditating, and dancing are used to experience a state of bliss that brings one closer to union with God.
The bhakti movement of India uses ecstatic singing, chanting, and meditation to bring the devotee closer to God. In the case of bhakti, the devotee is trying to get closer to the Hindu Supreme Being. For the case of the Alvars of south India, God is identified as Krishna. Another key group are the Lingayats, devotees of the god Shiva.
In both cases the literature emphasizes poetry, even song, extolling the virtue of God. Often this poetry describes God in very personal terms, and uses these familiar terms to describe the devotee's emotions. The Sufi poet Jelaluddin Rumi described God as a lover, but one who was sometimes distant, increasing his longing. At other times he was drunk on God, in a high like one feels when in love. This sort of passionate emotion is also seen in the poetry of the Bhakti movement. Krishna is all around, but he cannot be seen; he is hiding; he seems too far away.
If we want to look at some of the major themes in both we might note all the kinds of feelings one sees in love songs: a sense of loneliness because one's lover is far away, a sense of anger or upset or even jealousy that God is not easily understood or sometimes seems to disappoint; a sense of intoxication or bliss when one is filled with a sense of God's presence.
One more similarity: both rely on metaphor to describe the complex relationship the devotee has with God. The Tamil bhakti literature uses metaphors of scenery and landscape to convey a mood. As mentioned above, intoxication is a key metaphor for Rumi. The Sufi poet Fariduddin Attar developed an allegory about the migration of birds across a landscape to describe the Sufi path.
Finally, both emphasize the point of surrender. God is all around one. To experience God, one must abandon oneself and be filled with God. This is perhaps the point of Sufi and bhakti literature.