I found this definition of karakia really interesting (from the online Maori dictionary):
1.(verb) (-tia) to recite ritual chants, say grace, pray, recite a prayer, chant. Nā, ka mahia e ngā tohunga ka unuhia ngā rito kōrari, ka karakiatia kia mōhiotia ai, ka mate ka ora rānei (M 2005 wh280). / Now, the tohunga plucked the centre shoots of the flax, and recited incantations over them to ascertain whether the result would be defeat or victory. 2.(noun) incantation, ritual chant, chant, intoned incantation – chants recited rapidly using traditional language, symbols and structures. The two most important symbols referred to in karakia are of sticks and food, while the two key actions are of loosing and binding. Individual karakia tend to follow a pattern: the first section invokes and designates the atua, the second expresses a loosening of a binding, and the final section is the action, the ordering of what is required, or a short statement expressing the completion of the action. The images used in karakia are from traditional narratives. There were karakia for all aspects of life, including for the major rituals, i.e. for the child, canoe, kūmara, war party and the dead. Karakia for minor rituals and single karakia include those for the weather, sickness, daily activities and for curses and overcoming curses. These enabled people to carry out their daily activities in union with the ancestors and the spiritual powers.
Hanga rawa anō tōna whare wānanga, hei akonga mō ngā tamariki ki ngā tini karakia, ki tō rātou atua tapu, te karakia mākutu, te karakia ātahu, te karakia wehe, te karakia taupō, te karakia hono i te iwi whati, te karakia whakahoki mākutu, te karakia patu i ngā tapu, te karakia i ngā kanohi pura, te karakia mō te raoa, te karakia mō te haere ki te whawhai, te karakia whakaara i ngā tapu, te karakia kia ua te rangi, te karakia kia mao te ua, te karakia kia whatitiri, kia rū te whenua, kia maroke ngā rākau, kia maroke te wai, kia ngaru te moana, kia rokia te moana, kia haere mai ngā ika taniwha o te moana, ngā ngārara o te tuawhenua, te karakia o te huamata, te pure o te hua mai o te tau, te karakia o te kawenga ki roto i te rua tāhuhu, i te rua kōpiha rānei, te karakia o te tomokanga ki te ngāherehere, o te whakaputanga mai rānei i ngā manu mate ki waho o te ngahere, te karakia o te whakaatahanga o te whare o te whakatuheratanga hoki o te whare, te karakia o te nehunga tūpāpaku, te karakia o te whanautanga tamariki, o te whakaputanga hoki ki waho i te whare kōhanga, o te tohinga rānei i te ingoa (TJ 20/6/1899:3). / He built his academy of learning to teach the children the many ritual chants, their sacred god, karakia for witchcraft, to bewitch, to divert affections, for ?ulcers, to mend broken bones, to counter witchcraft, to kill using tapu, for blindness, for choking, for going into battle, to lift tapu, for rain, for rain to cease, to cause lightning, to cause earthquakes, to make trees dry up, to dry up water, to make the sea rough, to calm the sea, to attract large fish of the ocean and insects of the land, karakia for planting, to lift the tapu on a harvest to ensure a plentiful crop, for storing crops in covered pits or pits, karakia for entering the forest or for bringing dead birds out of the forest, karakia for building and opening buildings, for burying the dead, or childbirth and for leaving the house for childbirth and of the naming ceremony. See also karakia haumanu, karakia kikokiko, karakia whati, karakia whakahorohoro. 3.(noun) prayer, grace, blessing, service, church service. Kāore he utu mō ngā tīkiti, kāore hoki he ohaoha i roto i ngā karakia (TTT 1/10/1923:16). / The tickets are free and there is no offertory in the service.
How does ‘karakia’ compare with ‘whakamoemiti’ exactly, though? I’m not sure this definition answers the question, but:
1. (verb) (-tia) to praise, express thanks.
Kei te whakamoemiti atu te manu meroiti nei ki ētahi o ana kaitautoko e whiu mai nei i ngā hua kāramuramu hei oranga mōna kia kaha ai tana korokī i runga i tōna pae (KO 15/5/1885:4). / This small bird is thanking some of its sponsers who have tossed some karamū berries as sustenance so that its call on its perch will be loud.
2. (noun) praise, thanks.