Mukti does not necessarily always mean Moksha. it depends on how the word is used in the sentence.
For example in the words like Rogamukti (freedom from disease), Rinamukti (freedom from debt) etc it does not mean Moksha at all but simply freedom or to get rid off.
But when used in that context then Mukti means Moksha. Then those are just two different words which are used to refer to the same thing.
For example, we have the following verse from Devi Mahatyam:
सर्वभूता यदा देवी भुक्तिमुक्तिप्रदायिनी ।
त्वं स्तुता स्तुतये
का वा भवन्तु परमोक्तयः ॥
sarvabhūtā yadā devī bhukti-mukti-pradāyinī ।
stutaye kā vā bhavantu paramoktayaḥ ॥
When you [who are] devi, are all of manifestation, are giver of
nourishment [in this world and] liberation [from this world], are the
praised one, [then] for [your] praising, what ever can be great
That is, devi, you are beyond my ability to find words to praise you.
greatest of the great words fall short of praising you. [but even then
let me try].
The word by word meaning is as follows:
sarvabhūtā = all (sarva) manifestation, people, world ( bhūta) yadā =
when devī = goddess bhukti = consumption (food, needs etc.)
mukti = liberation (of soul from bondage) pradāyinī = giver (fem.)
tvaṃ = you [are] stutā = [THE] praised one stutaye = for praising
[you] kā = what vā = or, ever, (to indicate what ever could be ..)
bhavantu = can be paramoktayaḥ = greatest (parama) utterances
So, here Mukti is used to mean Moksha.
Further, we have from the starting verses of the Phalasruti of the Durga Satanama Stotram, found in the Viswasara Tantram:
Ya idam prapathen nityam durganamasatashtakam |
vidhyate devi trishu lokeshu parvati ||
Dhanam dhanyam sutam jayam
hayam hastinameva cha |
Chaturvargam tatha chante labhen muktincha
The meaning of these verses is "One who daily recites this Stotra comprised of hundred names of Goddess Durga there is nothing in the three worlds that he can not achieve. Also he gets the worldly pleasures like wealth, crops/food, wife, son etc. He obtains the Chaturvarga (i.e Artha, Kama, Dharma etc) and at the end obtains eternal Mukti."
Here again Mukti means Moksha.
Yet another verse from Kularnava Tantram's third chapter:
Salokyapramukham devi labhenmuktim chaturvidham |
sandehah sadhaka kulanayike ||
This verse is describing the result that one obtains by chanting a particular Mantra for four hundred times. It says that such an aspirant gets four kinds of Muktis or Moksha.
So, here too Mukti means Moksha.
(Four kinds of Muktis are Salokya, Sarupya, Sarshti and Sayujya )
There can be plenty of such references. Here are a few more from the Yogic text called Hatha Yoga Pradipika.
Sarveshāmeva bandhānām uttamo hyuddīyānakah
Uddiyāne drdhe bandhe
muktih svābhāvikī bhavet||
Of all the Bandhas, Uddiyâna is the best; for by binding it firmly
liberation comes spontaneously.
Sahajoliriyam proktā śraddheyā yoghibhih sadā
Ayam śubhakaro yogho
This is called Sahajolî, and should be relied on by Yogîs. It does
good and gives moksa.
So, in such contexts Mukti, Moksha, Kaivalya etc are all synonyms.