The literal meaning of the word Manmukh is such person whose mukh i.e. face is towards the mind i.e. one who follows the mind is a Manmukh. Bemukh is such person who turns his mukh or face away from the Guru. One who disowns his Guru is a Bemukh. The closest English word I can think of is an Apostate. The literal meaning of Bemukh could also be such person who does not have a mukh i.e. face. "Be" in Farsi means "without" and Mukh means face. A person who does not have mukh is such person who has no respect. A person who becomes a bemukh also becomes without face i.e. he loses respect in this world and the next one.
Traditionally, Manmukh is such person who has not yet become a Gurmukh i.e. has not gotten initiated into Sikhi. On the other hand, Bemukh is such person who was once a Gurmukh but has turned away from the Guru. In Siri Anand Sahib, Guru Sahib has described the state of a Bemukh as follows:
ਜੇ ਕੋ ਗੁਰ ਤੇ ਵੇਮੁਖੁ ਹੋਵੈ ਬਿਨੁ ਸਤਿਗੁਰ ਮੁਕਤਿ ਨ ਪਾਵੈ ॥
ਪਾਵੈ ਮੁਕਤਿ ਨ ਹੋਰ ਥੈ ਕੋਈ ਪੁਛਹੁ ਬਿਬੇਕੀਆ ਜਾਏ ॥
ਅਨੇਕ ਜੂਨੀ ਭਰਮਿ ਆਵੈ ਵਿਣੁ ਸਤਿਗੁਰ ਮੁਕਤਿ ਨ ਪਾਏ ॥
ਫਿਰਿ ਮੁਕਤਿ ਪਾਏ ਲਾਗਿ ਚਰਣੀ ਸਤਿਗੁਰੂ ਸਬਦੁ ਸੁਣਾਏ ॥
ਕਹੈ ਨਾਨਕੁ ਵੀਚਾਰਿ ਦੇਖਹੁ ਵਿਣੁ ਸਤਿਗੁਰ ਮੁਕਤਿ ਨ ਪਾਏ ॥੨੨॥Translation: If someone becomes Vemukh (turns away from the Guru) from Guru, such person can never attain salvation. You may go and ask wise people and they too will say the same thing i.e. one cannot get salvation from anywhere else. Such person wanders in many life forms but does not get emancipation without the true Guru. Then when he seeks refuge in the true Guru, the true Guru enables him to listen to the Shabda (i.e. Naam). Siri Guru jee using their Mohar (Nanak) asks us to think deeply that without the true Guru, one cannot get emancipation.
A person who becomes a Bemukh has to come back to the Guru and earn the true spiritual earnings of Naam. There is no other way out.
If we really think of it, a Bemukh is also a Manmukh because a person turns away from the true Guru only when he listens to his mind instead of Gurmat (Guru's teachings). So we can say that every Bemukh is also a Manmukh but every Manmukh is not necessarily a Bemukh. Now if we think more deeply, even us who are initiated Sikhs (Amritdhari) of Guru Sahib, are manmukhs up to some extent. A 100% Gurmukh is such person who follows 100% Gurmat i.e. is totally imbued in Naam. Such person is one with Vaheguru. Up until our minds are embroiled in Maya, we are Manmukh in proportion we are entangled in Maya.
Who is worse? It seems like a Bemukh is worse because he was once a Gurmukh but turned away from Guru Sahib. He gets heavier punishment than a Manmukh. Having said that, I think a Bemukh has a better chance at attaining salvation as compared to a Manmukh because since he once received the Gurmantra / Naam from Guru Sahib, it will definitely sprout one day. It may take time but a Bemukh is surely destined to be Mukat (emancipated) one day. Manmukh may wander till infinite time. The above stated pauri from Siri Anand Sahib too gives hope to Bemukhs by saying that they will become free when they come to the refuge of the true Guru.
As far as two kinds of spellings - Bemukh or Vemukh - are concerned, both are correct and have been used in Gurbani. Most common use is Vemukh but Bemukh too is used as is evident from two pankitis listed below.
ਸਾਕਤ ਨਰ ਸਤਿਗੁਰੁ ਨਹੀ ਕੀਆ ਤੇ ਬੇਮੁਖ ਹਰਿ ਭਰਮਾਵੈਗੋ ॥
ਜਨ ਕਾ ਰਾਖਾ ਆਪਿ ਸੁਆਮੀ ਬੇਮੁਖ ਕਉ ਆਇ ਪਹੂਚੀ ਮੀਚ ॥1॥ ਰਹਾਉ ॥Kulbir Singh
ਸਤਿਗੁਰਬਚਨਕਮਾਵਣੇਸਚਾਏਹੁਵੀਚਾਰੁ॥
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