The two words seem quite similar in their meaning.
Synonyms: acquiescent, agreeable, amiable, duteous, dutiful, obliging, placable; soft, surrendering, yielding; fawning, kowtowing, obeisant, obsequious, servile, slavish, subordinate, subservient (Merriam-Webster)
Synonyms: submissive, deferential, acquiescent, compliant, accommodating, obedient, dutiful, duteous, biddable, yielding, meek, docile, ductile, pliant, passive, unassertive, spiritless, subdued, humble, timid, mild, lamblike (Oxford)
They both mention the word ‘acquiescent‘ as their synonym, yet neither word had a mention of the other as its synonym. I am confused as to why this is so.
There is much overlap between describing someone as complaisant, subservient, compliant or acquiescent.
The Oxford definition of acquiescent is
ready to accept something without protest or do what someone else wants.
The root of acquiescent is quiet. If someone is acquiescent he or she is not does not raise objections when asked to do something, or when inconvenienced in some way.
The other three words (complaisant, subservient and compliant) are all partial synonyms of acquiescent, but differ in motive and degree.
For complaisant, (which should not be confused with complacent), the Oxford definition begins
willing to please
and if we say someone is complaisant we mean he is keen to please others and go along with their wishes. A complaisant friend, when discussing a proposed day out, does not much mind where to meet or go, just as long as everybody else is happy. The Oxford definition gives an example of complaisant doctors signing sick notes, even for healthy patients, just because they want to please. However, just because people are complaisant does not mean they can be ordered about or treated as servants.
For subservient the definition begins
prepared to obey unquestioningly.
A person is subservient (to another person) if he does exactly what the other person tells him. Usually one would be subservient only to somebody of a superior status, but to be described as subservient would be even more obedient and respectful than usual, given the status difference. The example given is of someone subservient to her parents. Unlike a complaisant person, a subservient person may not particularly want to please his superior, although probably worries about the consequences of displeasing him. The main idea is obedience, whether from fear, respect or duty, or to curry favour. A subservient person accepts being treated as a servant. (If currying favour with a boss then both complaisance and subservience may apply.)
For compliant the definition begins
disposed to agree with others or obey rules
A compliant person tends to do what is expected or reasonably required from him, because that is his disposition. We may say that a car is compliant regulation X, meaning it meets that rule.