ANSWER: Both mean
help, but 우 (佑) seems to refer to
help from above while 좌 (佐) seems to refer to
help from below.
The Chinese character 右 (우) means
right or
right hand, and the Chinese character 左 (좌) means
left or
left hand. And the character 人 (인) or 亻(인) means
person. While 人 is normally used by itself, the character 亻(인) is used only as a component of composite characters, such as in the characters 佑 (우) and 佐 (좌). Therefore, the components of 佑 and 佐 literally translate as "a person's (亻) right hand (右)" and "a person's (亻) left hand (左)."
The phrase 천우신조 (天佑神助) can translate as
divine care or
grace of God, so the phrase 천우신조로 translates as
by the grace of God, but the phrase 천우신조 (天佑神助) literally translates as "Heaven's (天) help (佑) [and/or] God's (神) help (助)." When referring to
help from Heaven, notice that 우 (佑) was used, not 좌 (佐)
The first sentence of Korea's national anthem reads as follows:
동해(東海)물과 백두산(白頭山)이 마르고 닳도록 하느님이 보우(保佑)하사 우리 나라 만세(萬歲).
Until the waters of the East Sea (동해물) dry up (마르고) and Mount Baekdu (백두산) wears away (닳도록), God (하느님이) protect (보 保) and help (우 佑) our country (우리 나라) ten thousand years (만세).
Notice that the subject 하느님 (God) was used with the verb 보우(保佑)하다, which means
to protect (保)
and help (佑). Again, when the subject is
God, the character 佑 (우) is used, not 佐 (좌). In addition, the character 우 (佑) or the verb 보우(保佑)하다 seems to have been used to refer not only to
help from God but also to
help from kings and emperors. This is why I wrote that 佑 (우) seems to refer to
help from above.
As for the character 佐 (좌), it is used in the verb 보좌(保佐)하다, which also means
to protect (保)
and help (佐), but it refers
to giving protection and help to a superior. 보좌관 (補佐官), for example, translates as
an aide, and 보좌인 (保佐人) as
an assistant,
a counselor, or
an advisor, suggesting that the character 佐 (좌) is used when referring to
help from below.
In summary, one seems to
get help (우 佑) from God and king but
gives help (좌 佐) to one's king or superiors. In other words,
help from a person's right hand (佑), and
help to a person's left hand (佐).