Literally the word Avadhuta means "one who has shaken off" the worldly existence. So, it denotes a free, liberated being. And, Jivanmukta means the same too viz: "one who has attained liberation while still living".
Various kinds of liberation are mentioned like Jivanmukti (which is achieved while living), Videhamukti (which is achieved not while living but after death) and Kramamukti (which will happen for sure sometime in the future but not certain exactly when).
Therefore, there is basically no difference between an Avadhuta and a Jivanmukta. Both has achieved the knowledge that liberates and they have done so while still living.
Lord Shiva gives the following definition of an Avadhuta which further proves my point:
aksharatvAd-varenyatvAd-dhuta-samsAra-vandhanAt |
tattvamasya-artha-siddhatvAt avadutohabhidhiyate ||Since he is immutable/imperishable (akshara), best among/most excellent (varenya), since for him the worldly bondage are all removed (dhuta-samsAra-vandanat) and since he has realized the meaning of "Tvat Tavam asi" (you are that) -- he is called Avadhuta.
KulArnava Tantram 17.24
Now, Avadhuta is also the name of a class of Sanyasis (ascetic) as mentioned in some minor Upanishads:
[174] "There are six types of renunciation: Kuticaka, Bahudaka, Hamsa, Paramahamsa, Turiyatita, and Avadhuta.55 A Kutlcaka has a topknot and a sacrificial string, carries a staff and a water pot, and wears a loincloth and a patched garment. He is devoted to honoring his father, mother, and teacher, to the use of articles56 such as a pot, a spade, and a sling [LSU 21, n. 27], and to the practice of eating food obtained from one place. He wears a white vertical mark on his forehead and carries a triple staff.
From the Narada Parivrajaka Upanishad
Among these types, the Avadhuta and Turiyatita are considered as the highest kind and they are considered as free while living or in other words as Jivanmuktas.
"To those who renounce at a time of mortal danger and to Kuticakas belong the earthly world and the atmospheric world, respectively.
To Bahudakas belongs the heavenly world, to Hamsas the Penance-world, and to Paramahamsas the Truthworld [ArU 4, n. 3]. [178] To Tunyatitas and to Avadhutas belongs the attainment of liberation in their very selvesTo Tunyatitas and to Avadhutas belongs the attainment of liberation in their very selves through the deep meditation on their own true nature in the manner of the wasp and the worm
We also have the Avadhuta Gita where the Avadhuta refers to Lord Dattatreya who is the best example of a Jivanmukta.
So, although two different words, both refer to a "liberated while living being".