Mattan Klein was born and raised in Jerusalem, the son of Prof. Michael Klein, dean of Hebrew Union College's Jerusalem institute for Jewish studies, and a mother who taught music for Israel's Ministry of Education. He started on piano at seven, switched to flute soon after, and by his early teens was touring Europe and North America with Israeli folklore groups. At 15 he decided jazz would be his focus, and at 16 he won first prize in the Pizmon-Tichon composition competition setting music to lyrics by the poet Natan Alterman. Toward the end of his regular IDF service in 1994 he transferred into the education branch associated with Unit 8200 and performed with its band.

He studied at Jerusalem's Rubin Academy of Music and Dance from 1995, then moved to Boston in 1996 to finish a degree at Berklee College of Music. A year later he performed in Boston alongside Berklee faculty including vibraphonist Victor Mendoza, who brought him to Tel Aviv's Camelot club for seven shows. From 1998 to 2008 Klein lived in New York, composing and recording, winning awards in the John Lennon Songwriting Contest and the USA Songwriting Competition, and performing at Carnegie Hall and the Kennedy Center. He also recorded flute sample libraries during this period, including a Sony Native Instruments product and Orange Tree Samples' "Passion Flute," and backed visiting Israeli artists such as Miki Gavrielov and Yoni Rechter on their US dates.

Klein returned to Israel in 2008 and founded Four Flute Flight, a four-flute-plus-rhythm-section ensemble, with grant support from Israel's Ministry of Culture and Sport. He recorded a further New York album in 2011 with flutists Itai Chris and Hadar Neuberg, then in 2012 released Naomi's Groove, world-music arrangements of songs by Naomi Shemer featuring Amos Hoffman, Ofer Ginor, Gilad Dovrat, Roy Oliel, Tom Oren and Ella Tadmor. From 2013 he became a regular accompanist for Miki Gavrielov, a relationship that continued through a 2024 concert marking Gavrielov's 60-year songwriting career with the Tel Aviv Philharmonic. Between 2010 and 2019 he also served as music director and arranger for the Hof HaSharon Regional Council's "Culture Heroes of the Sharon Coast" concert series, working with a roster of major Israeli songwriters including Shlomo Gronich, Yoni Rechter, Shem Tov Levy, Efraim Shamir, Alon Oleartchik, Miki Gavrielov, Nurit Galron and Yitzhak Klepter. He now lives in Tel Mond, in the Sharon region.

His 2016 album Sound Tracks coincided with a solo turn at the London Jazz Festival with an international flute big band and a performance in Montreal's Power Jazz Series. He represented Israel at the Jazz Education Network conference in New Orleans in both 2017 and 2020, led a flute ensemble at Thelma Yellin High School for Performing Arts, and took his group to Ghana's Israel Week in Accra in 2017. His 2019 album Wag the Mammoth in the China Store, distributed by Helicon and backed by Israel's ACUM performance-rights society, drew strong reviews and led to a festival date in Tirana, Albania. In 2021 he founded and became artistic director of the Kfar Yona Jazz Festival, and in 2022 released his Ubuntu Music debut The Long Run, an homage to Brazilian composer Hermeto Pascoal recorded with his regular quartet plus guest guitarist Nitzan Bar. That year also brought dates at Lithuania's Klaipeda International Jazz Festival, the Cambridge Jazz Festival in England, Serbia's Nisville Jazz Festival, and Tel Aviv's Summertime Jazzfest.

In 2023 Klein became artistic director of the first Tel Aviv International Flute Festival, held at Shablul Jazz club, and formed the jazz-funk project Spice'n'Space with drummer Uria Mader; their debut album The Funk Thing followed on Ubuntu Music with a supporting European tour. In 2025 he produced and played flute on Countryside Motorways by the group ZurHub for the US label Origin Records, alongside bassist Assaf Hakimi.

Klein's music runs on a consistent cross-cultural axis: Israeli, Brazilian, Middle Eastern and, more recently, funk idioms, filtered through a flute sound reviewers describe as clear, bright and melodic. The Long Run is a direct tribute to Hermeto Pascoal, and critics have connected its long melodic lines and modulated rhythms to 1970s jazz fusion in the vein of Chick Corea's Return to Forever. The Funk Thing, made as Spice'n'Space with Uria Mader, points the same instrument at old-school funk and soul-jazz touchstones: The J.B.'s, Lou Donaldson, Grant Green, Herbie Hancock's harmonic language and the contemporary groove band Vulfpeck. Klein has summed up his own outlook simply: "There's music from everywhere in Israel, and you just have to open your ears." Early in his career, just after military service, he led a group he punningly named Mas u'Mattan (a play on his own first name and the Hebrew phrase for "negotiation" or "give and take"), a small sign of the same restless, absorb-everything approach that later produced Naomi's Groove's world-music arrangements of Naomi Shemer.