No, the terms are not synonymous. A poem is a composition in verse. A palindrome is a series of characters that can be read the same forward or backward. (e.g. words such as racecar, or phrases such as 'Able was I ere I saw Elba'.) It is possible for a poem to be made up of palindromes, but it is not common.
5 lines in a limerick (type of poem...)
"Radar" is a palindrome for a type of flower.
Minim is a palindrome for a type of music note
kayak
A mum is a flower that is also a palindrome.
Pop is a type of music.
Limerick poetry is named after the city of Limerick in Ireland. It consists of five lines, with a distinctive rhyme scheme and rhythm.
kayak
Civic
poppop
mum
pop