Both of these poems reflect on the nature of death, but they differ in many ways.
"Death, be not proud" by John Donne is actually "Sonnet X" of the poet's Holy Sonnets, and thus part of a sequence of poems on related topics as opposed to a fully independent lyric poem. It was written in 1609, and is an overtly religious poem, beginning with a description of the might and power of death, and then, in the sestet, following the notion expressed in 1 Corinthians 15:26 that "The last enemy that shall be destroyed is death", and arguing that death in this world is the beginning of eternal life.
Alfred, Lord Tennyson (1809–1892) requested that "Crossing the Bar" be placed as the final poem in all collections of his work. Written in 1889, this poem also portrays death optimistically, as a transition rather than an ending, but is less overtly religious than Donne's work and not specifically Christian.