As Patchunky mentions, both are common services in the Eastern Christian Churches of the Byzantine-Constantinopolitan tradition.
The Moleben [Service of Supplication] is essentially a condensed form of Matins originating with the ancient Pannychis and has essentially four general categories - festal [commemorating a particular feast or saints on their feastdays, for festal times such as Pascha, etc.], devotional [to a certain wonderworking icon of the Mother of God, for example, or to the Savior, Mother of God, St. Nicholas, the patron saint of the temple, etc. at any time], supplicatory [sung in times of dire need as for the sick, natural disasters, etc.], and for specific needs [taking abode in a new house, lesser blessing of water, thanksgiving, etc.]. There is also a variant [Parastas] that is used for the commemoration of the dead as well.
The Moleben is a very common and extremely well-loved service especially amongst the Ukrainian, Carpatho-Rusyn and Russian Churches.
This service is also known amongst Churches of the Greek liturgical tradition as the paraklesis, but it is celebrated much less often.
In many Ukrainian parishes, for example, rather than having “mass” celebrated for an intention the petitioner may rather ask the priest to celebrate a Moleben for the intention.
The Akathist [literally “without sitting” referring to the prayer posture while praying one] is a poetic composition of verses to Christ, the Mother of God, a wonderworking icon, a Saint or a specific feast. The original Akathist is that to the Mother of God generally attributed to St. Roman the Melodist in the 6th century. It was originally intended to be a kontakion or a proper hymn to the Mother of God for use within the liturgical services of the Feast of the Annunciation but became quickly popular as a paraliturgical service taken at various times outside of the actual feast day.
The original Akathist to the Mother of God was the first known specific paraliturgical hymn or service to the Mother of God predating the Rosary as known in the Latin Church by six or more centuries.
In the Greek Church, the original Akathist to the Mother of God is the only one usually celebrated, and that during Great Lent. But amongst especially the Ukrainians and Russians there are hundreds of Akathists to Christ, the Mother of God, saints, wonderworking icons, feast days, thanksgiving, for Holy Communion, for the dead, etc.
The Akathist can be celebrated within a Moleben, as part of Compline, within Matins [such as for Akathistos Saturday] and is often celebrated essentially alone with beginning prayers, sometimes Psalm 50 and the Creed, etc.
FDRLB