Of these four synonyms, or expressions with similar meanings, fault is probably the most widely term used.
Fault is not so much used to talk about someone's character, Helen. Instead we talk about electrical, mechanical or technical faults:
A fault then describes a weakness in something, primarily. But sometimes it is used to describe a weakness in someone's character:
There was a fault in the wiring and I had no idea how to correct it.
There was a delay in the broadcast of the programme and this was due to a technical fault.
A mechanical fault caused the train to come off the rails.
We also have the frequently used expression: It's (not) my/your/his/etc fault. This is a more idiomatic way of saying: I am (not) to blame or I am (not) responsible (for this unfortunate situation).
She has her faults, but, on the whole, she's a nice person.
We all have our own faults, I suppose.
It's not my fault he's late. Don't blame me.
I'm sorry. It's my fault. I forgot to pass on the message.
If you don't get enough sleep, it's entirely your own fault.
It was partly the teacher's fault for giving them too much homework.
We use flaw mainly to talk about a minor fault or weakness in something which make it less effective or valuable. We talk about flawed arguments for example. Note also a flawless complexion:
However, we can also talk about serious or major flaws:
There's a flaw in your argument. I agree with you up to a point, but the last part doesn't make complete sense to me.
There was a tiny flaw in the necklace and it certainly wasn't worth all the money we had paid for it.
She attributed her flawless complexion to the moisturising creams she used.
And, yes, we can also use flaw to describe a fault in someone's character:
There are major flaws in the way we train teachers in this country.
There were serious flaws in the construction of the pedestrian bridge.
The only flaw in his character was his short temper ?he tended to fly off the handle at the slightest provocation.
Weaknesses generally describe the state or condition of being weak and of lacking strength or resilience.
Weaknesses can also refer to faults or problems that make something less attractive or effective:
The main weakness of this government is that it keeps changing direction on key policy issues.
He showed great weakness in not owning up to his part in the bad behaviour.
Note that if you have a weakness for something, you are very fond of it:
They were keen to know how well it would sell in Russia so they listed all the strengths and weaknesses of their product for this market.
The only weakness in her character that I could spot was that she seemed to be over-dependent on others.
I have a great weakness for chocolate. I can never refuse it.
We use drawback to refer to a feature of something which makes it less useful or acceptable than it could be. Drawback is often synonymous with disadvantage, but note that drawforward does not exist as an alternative to advantage!
So, Helen, from the shades of meaning inherent in all four of these terms, note that we can refer to faults, flaws and weaknesses in someone's character, but we are less likely to talk about drawbacks in someone's character.
The only drawback / disadvantage with this accommodation is that it's a fifteen-minute walk to the bus-stop.
The main drawback of this examination is that it takes two months before the results are released.