Both narrow what you are responsible for, but for opposite ends. Stoicism frees your effort in order to spend it better — a discipline of will. Taoism subtracts until the deliberateness is gone — a suspicion of will. One tidies the aiming; the other removes it.
The overlap, which is real
Put Epictetus beside Laozi and the agreement is immediate. Both begin by narrowing what you are responsible for. Both treat the struggle against circumstance as the source of most trouble. This is why the two get shelved together, and why a reader who arrives through one usually ends up reading the other.
Of things some are in our power, and others are not. In our power are opinion (hupolaepsis), movement towards a thing (hormae), desire, aversion (echchlisis), turning from a thing; and in a word, whatever are our acts. Not in our power are the body, property, reputation, offices (magisterial power), and in a word, whatever are not our own acts. And the things in our power are by nature free, not subject to restraint or hindrance; but the things not in our power are weak, slavish, subject to restraint, in the power of others. Remember then, that if you think the things which are by nature slavish to be free, and the things which are in the power of others to be your own, you will be hindered, you will lament, you will be disturbed, you will blame both gods and men; but if you think that only which is your own to be your own, and if you think that what is another’s, as it really is, belongs to another, no man will ever compel you, no man will hinder you, you will never blame any man, you will accuse no man, you will do nothing involuntarily (against your will), no man will harm you, you will have no enemy, for you will not suffer any harm.
He diminishes it and again diminishes it, till he arrives at doing nothing (on purpose). Having arrived at this point of non-action, there is nothing which he does not do.
The point where they part
Read in sequence, the difference shows. Epictetus divides the world so that you can act correctly inside your half; his verbs are all about effort properly aimed. Laozi subtracts until the deliberateness is gone, and then says nothing is left undone.
One is a discipline of will. The other is a suspicion of will. The Stoic frees up effort in order to spend it better; the Taoist frees it up in order to stop spending it. That is the whole distance between them — smaller than it looks and more consequential than it sounds.
It shows most clearly in what each does with a goal. Epictetus keeps the goal and detaches from the outcome. Laozi is suspicious of the goal itself.
He who acts (with an ulterior purpose) does harm; he who takes hold of a thing (in the same way) loses his hold. The sage does not act (so), and therefore does no harm; he does not lay hold (so), and therefore does not lose his hold. (But) people in their conduct of affairs are constantly ruining them when they are on the eve of success. If they were careful at the end, as (they should be) at the beginning, they would not so ruin them.
Both say the trouble is in the judgement
The agreement goes deeper than the dichotomy. Encheiridion 5 is the most quoted sentence in Stoicism, and it locates the problem precisely.
Men are disturbed not by the things which happen, but by the opinions about the things; for example, death is nothing terrible, for if it were it would have seemed so to Socrates; for the opinion about death that it is terrible, is the terrible thing. When then we are impeded, or disturbed, or grieved, let us never blame others, but ourselves—that is, our opinions. It is the act of an ill-instructed man to blame others for his own bad condition; it is the act of one who has begun to be instructed, to lay the blame on himself; and of one whose instruction is completed, neither to blame another, nor himself.
Men are disturbed not by the things which happen, but by the opinions about the things. Marcus Aurelius states the same thing as an instruction rather than a diagnosis.
Take away thy opinion, and then there is taken away the complaint, "I have been harmed." Take away the complaint, "I have been harmed," and the harm is taken away.
Take away thy opinion, and the complaint goes with it. This is a repair procedure: find the judgement, remove it, and the disturbance has nothing to stand on. Zhuangzi arrives near the same place and then keeps walking.
Where Zhuangzi goes further than the Stoics
The Stoic corrects a judgement by replacing it with a true one — look at things as they are in truth, as Marcus puts it. That assumes there is a correct view available to a person who works at it. Zhuangzi doubts the assumption.
Those who dream of (the pleasures of) drinking may in the morning wail and weep; those who dream of wailing and weeping may in the morning be going out to hunt. When they were dreaming they did not know it was a dream; in their dream they may even have tried to interpret it; but when they awoke they knew that it was a dream.
In their dream they may even have tried to interpret it. That sentence is aimed at exactly the Stoic move. Examining your impressions is itself an impression; the correction is made from inside the position being corrected. Epictetus never entertains this worry, and Zhuangzi never lets go of it.
So the two traditions share a diagnosis and split on the remedy. The Stoic trusts the examined judgement and builds a discipline on it. The Taoist suspects that trust, and the response is to hold judgements loosely rather than to perfect them.
A difference of temperature
Epictetus is a teacher with a classroom voice — he anticipates your objection and answers it, often impatiently. The Tao Te Ching does not argue. It states, and lets the statement sit, and if you disagree it offers nothing further.
This changes how each book is used. The Encheiridion rewards being consulted: arrive with a situation and it hands you a distinction. The Tao Te Ching resists that. Consulted for advice it gives back something that sounds like a riddle, and readers expecting the Stoic transaction usually decide it is vague. It is not vague — it is describing a disposition rather than prescribing a response.
Death, where the contrast is sharpest
Both traditions talk about death constantly, and the tone could not be more different. Marcus writes it as pressure — a deadline that should change what you do today.
Do not act as if thou wert going to live ten thousand years. Death hangs over thee. While thou livest, while it is in thy power, be good.
Death hangs over thee. While thou livest, while it is in thy power, be good. The urgency is the argument: mortality is a reason to act well now, and the sentence is built to be uncomfortable.
Zhuangzi treats the same fact as a reason to stop straining. Chapter 2 arrives at death through the dream passage, and the effect is to loosen the grip rather than tighten it — if waking and dreaming are hard to tell apart from inside, so is living and whatever follows, and the anxiety loses its object.
Neither is consoling in the ordinary way. But one uses death to make you serious and the other uses it to make you lighter, and that difference reaches back into everything else the two books recommend.
Where each locates the problem
For Epictetus the trouble is a judgement inside you, and the correction is also inside you: revise the judgement and the disturbance goes. For Laozi the trouble is more often the wanting itself, and the correction is subtraction rather than revision.
Neither position is obviously right, and the traditions were never in conversation — they are separated by continents and centuries. What makes the comparison useful is not influence but contrast: two answers to the same complaint, arrived at independently.
Where to start, from either side
Coming from Stoicism, the least jarring entry to the Tao Te Ching is chapter 8, which reads almost as a Stoic passage about status. Coming from Taoism, the Encheiridion's first chapter is the shortest statement of the shared ground. Neither book asks you to leave the other.
Both are on this site in full: the Encheiridion in George Long's translation, the Tao Te Ching in James Legge's. Neither has been paraphrased or modernised.