The first one is used in ordinary Japa and in Archana (Flower offered with calling names of Krishna).
Second one is used in Tantra Sastra where the japa is intensive and requires a Guru for getting Mantra diksha. Mantra-Diksha (initiation) being a most sacred ritual, should not be taken lightly. The aspirant should not get himself initiated just to imitate others.
Tantrik worship gives quicker and better results.
1) 'Om Shri Krishnaya Namaha' is the first name in Krishna Asthotram.
108 Names Of Lord Krishna ( Krishna Ashtotram)
Om Shri Krishnaya Namaha
Om Kamala Nathaya Namaha
Om Vaasudevaya Namaha
Om Sanatanaya Namaha
Om Vasudevaya Namaha
.............................................
Lord Krishna was Lord Vishnu's eighth incarnation and this is His 58th name as listed in the Vishnu Sahasranama, the thousand names of Lord Vishnu. Krishna means black or dark-coloured. Lord Krishna's complexion was said to be dark like the most beautiful clouds full of rain. This means that Lord Krishna is full of sacred grace and blessings and He is always ready to shower this auspicious energy on His true devotees.
2) 'Om Kleem Krishnaya namah'
The mantra of lord Krishna, the incarnation of Lord Vishnu. Solves all the problems of devotee and bestows him with wealth and prosperity. Lord Krishna gives Artha, KAMA and moksha. For Gopis and Meera all the Artha, KAMA and Moksha were Lord Krishna.
3) 'Om Kleem Krishnaaya Govindhaya Gobi Jana Vallabaaya Namaha/swaaha' (Namaha for Japam and Swaaha for homam)
In making an offering to Krishna, the mantra that is used is Kleem Krishnaaya Govindaaya Gopijanavallabhaaya Swaaha. It is interesting to explore the full significance of this mantra. Kleem means the earth. It is the first element. Krishnaaya signifies water. 'Govindaaya signifies Fire (Agni). 'Gopijanavallabhaaya' signifies Air (Vaayu). 'Swaaha' signifies Space (Akaasa). The mantra thus represents all the five elements in the cosmos as manifestations of the Divine. The Cosmos cannot exist without these five elements. They are all pervading. It follows that there is no place in the Cosmos where God is not immanent.
The GOPIKAS, through this mantra, taught the world that God should be adored as the omnipresent Divine everywhere in the Cosmos. Hence in every atom, in every particle, the Divine is present all the time.
AUM (OM) is the Bija Mantra, from which all mantras and other Mantras originated.
KLEEM:
This is the Kama bija. Ka means the Lord of desire, Kamadeva, for those aspire materialistic desires. Ka here means Krishna for those aspire Lord Krishna. La means Indra. Ee means contentment or satisfaction.Terminal M is ChandraBindu (Nada and Bindu). This means that which brings happiness.
Mantras are syllables, phonemes, quasi-morpheme and its series, morphemes, single word, words, phrases, sentences, verses, and passages. Examples of Phonemes are hreem, hram, and hrum ending in 'm', which can be intonated as long as one's breath can sustain it. HREEM: H = Siva, R = Sakti Prakrti, I = Mahamaya and Terminal M is Chandrabindu. Chandrabindu = Nada and Bindu. Nada is Progenetrix of the Universe and Bindu is Brahman as Isvara and Isvari (Isvaratattva). There is another interpretation HREEM: H = gross body; R = subtle body; I = causal body and M = Turiya state. Sakti is the causal body of the subtle and gross bodies of all living beings.
HREEM (Devi): H =Siva, R = Sakti, I = Mahamaya and M = Chandrabindu
SREEM (Lakshmi): S = Mahalaksmi, R = Wealth, I = Satisfaction and M = Chandrabindu
KREEM (Kali): K =Kali, R = Brahma (Siva), I = Mahamaya and M = Chandrabindu
DUM (Durga): D = Durga, U = protection and M = Chandrabindu
Bija Mantras are the seed (based on Sanskrit alphabet) mantras, like Lam the seed sound of Muladhara Chakra, Vam of Svadhisthana, Ram of Manipura, Yam of Anahata, Ham of Visudha, and Om of Ajna. The first is Maya Bija, the second Lakshmi Bija, the third Kali Bija, the fourth Kurca Bija, the fifth Varma Bija, the sixth Astra Bija. Ram is Agni Bija, Em is Yoni Bija, Kleem is Kama Bija, Sreem is Badhu Bija, Aim is Sarasvati Bija, Kreem is Kali Bija, Hreem is the Maya bija and so forth. The first letter of the devata serves as the Bija Mantra: Gam for Ganesa, Dum for Durga.--
('Shakti and Shâkta' by Arthur Avalon alias Sir John Woodroffe, 1865-1936, Chapter Twenty-four 'Shakti as Mantra (Mantramayi Shakti), Book, Personal Collection).
Edited:
Respected Deito Sir,
I agree that there is no big or small mantra for a perticular deity.
Mantra (Om Krishnaya namah) with simple recitation is recommended for ordinary people.
If anybody is in a higher level, the same basic mantra with one or more of the following can be recommended by a guru, depending upon the nature of the disciple.
Tantrik worship (which gives quicker and better results):
Mantra is ritualistically followed with the anganyasa, karanyasa, mudra, avahana, pratisthana, offerings, rituals with yantra and with the prescribed bija mantra(s).