"Adon Olam" and "Yigdal" are widely used Jewish hymns, sung at the conclusion of prayer services. Most synagogues use one or the other, and it's not wrong to choose only one or recite both. My own synagogue that I previously attended usually sang "Adon Olam", although they occasionally sang "Yigdal". I preferred the latter. I liked its strong, assertive melody, and the repetitive syllables made it easier to remember and recite. While I'm now secular, due to life's circumstances that are beyond the scope of this thread, I still fondly remember these hymns.
"Adon Olam" has a slower, more plaintive melody, and is basically a poem set to music. It praises God while describing his beauty and power. While the tunes vary widely, from the original to contemporary pop songs, my former synagogue used a slow melody, with occasional lines sung in a faster, more staccato rhythm. I'm aware that it's the more preferred one of the two.
https://en.wikipedia.org/wiki/Adon_Olam
"Yigdal" is usually sung in a major key; it lists Maimonides's 13 creeds of Judaism, and the words are written in strict, formal tone. When my synagogue sang this one, they used an assertive-sounding, almost march-like melody. I found it interesting it has a repetition of the "-toh" ending; a fellow congregation member explained that they're adjectives converted into nouns, equivalent to the "-ness" ending in English.
https://en.wikipedia.org/wiki/Yigdal