In the Upanishads, there are two metaphors that are both included. These are "The chariot" and "The two birds" but what is the difference between the two?
मन एव मनुष्याणां कारणं बन्धमोक्षयोः ।
Mind is the only cause of bondage and liberation - Maitrayini Upanishad 4.3.11
As we know that mind, by its nature easily gets deviated towards the Samsara and without gaining control over it, it's impossible to attain Brahman. So, we first need to know about Manas, Antahkarana, Indriyas, Tanmatras, Buddhi etc. elements separately and their association with Jiva. Thereafter we need to learn observation and control of mind. Samkhya helps in first stage and Yoga in later.
Both metaphors - "the chariot" and "two birds" examples encourages us to control our mind to make it concentrated on Paramatman.
Let's first look at the chariot example.
आत्मानँ रथितं विद्धि शरीरँ रथमेव तु ।
बुद्धिं तु सारथिं विद्धि मनः प्रग्रहमेव च ॥ ३ ॥
3. Know the atman as the lord of the chariot, the body as only the chariot, know also intelligence as the driver; know the minds as the reins.
4. The senses, they say, are the horses; the objects which they perceive, the way; the atman, the senses and the mind combined, the intelligent call the enjoyer.
5. But of him who is not possessed of discrimination, and whose mind is always uncontrolled, the senses are not controllable as vicious horses of a driver.
6. But of him who knows and has a mind always controlled, the senses are always controllable as the good horses of the driver.
The message is very clear from the above self-explanatory verses of Katha Upanishad.
So, that was all about detaching the self from mind. Now, let's understand the relationship between Jiva and Ishwara with second metaphor of two birds.
Riveda 1.164.20 = Mundaka Upanishad 3.1.1 = Shvetashvatar Upanishad 4.6
द्वा सुपर्णा सयुजा सखाया समानं वृक्षं परिषस्वजाते ।
तयोरन्यः पिप्पलं स्वाद्वत्त्यनश्नन्नन्यो अभिचाकशीति ॥
IV-6: Two birds of beautiful plumage, who are inseparable friends, reside on the self-same tree. Of these, one eats the fruits of the tree with relish while the other looks on without eating.
IV-7: Sitting on the same tree the individual soul gets entangled and feels miserable, being deluded on account of his forgetting his divine nature. When he sees the other, the Lord of all, whom all devotees worship, and realizes that all greatness is His, then he is relieved of his misery.
Here body is represented by tree and Jiva (Individual self conditioned by ignorance or Maya) & Ishwara (True/Supreme self) are represented by two birds.
Jiva under the influence of ignorance tastes the fruit of Samsara, experience pleasure and grief and stays stuck in rebirth cycle. When Jiva worship, mediate and realize the Ishwara/Paramatma who is director of both Jiva and Samsara, s/he gets released from the grief.
This can be interpreted in two ways:
As per Vishishtadvaita, there lives two salves Jivatma and Paramatma within the body and Jivatma upon choosing Paramatma over Samsara (Mundaka Upanishad 3.2.3 = Katha Upanishad 1.2.23) and by the grace of Paramatma, attains the Moksha.
As per Advaita, when Jiva by practicing Yoga destroys the ignorance, realizes that I'm not body but I'm that (param)Atman and attains Moksha. And there is no other way to attain Moksha than knowing and realizing Brahman (Shukla Yajurveda 31.39 = Svetasvatara Upanishad 3.8). Visit this post for Advaita interpretation on this verse.
Both metaphors are complementary, not competitors. However as the difference between them is asked in the question, it can be said that:
Chariot example teaches us the detachment of our true self from subtle and gross body elements and Two Birds example teaches us to concentrate on immortal (Ishwara) rather than mortal (Samsara) to attain the ultimate reality.
Chariot metaphor preaches how to walk on the path of spirituality and Two Bird metaphor preaches where to go i.e the destination destination of your spiritual journey
The 2 verses are as follows.:
- Two Birds.:
द्वा सुपर्णा सयुजा सखाया समानं वृक्षं परिषस्वजाते । तयोरन्यः पिप्पलं स्वाद्वत्त्यनश्नन्नन्यो अभिचाकशीति ॥१॥
III-i-1: Two birds that are ever associated and have similar names, cling to the same tree. Of these, one eats the fruit of divergent tastes, and the other looks on without eating.
समाने वृक्षे पुरुषो निमग्नोऽनिशया शोचति मुह्यमानः । जुष्टं यदा पश्यत्यन्यमीशमस्य महिमानमिति वीतशोकः ॥२॥
III-i-2: On the same tree, the individual soul remains drowned (i.e. stuck), as it were; and so it moans, being worried by its impotence. When it sees thus the other, the adored Lord (Ishwara), and His glory, then it becomes liberated from sorrow.
Here the 2 birds are Jivatma (Individual Soul) and Paramatma (Universal Soul).
Fruits- Material Desires.
So, in simple word, the verse means, in the material world 2 Parameshvara and Jiva remains together, where Parmeshwara remains independent of Maya and just witnesses everything but jiva remains stuck in Maya and experience both happiness and sorrow again and again means stuck in the cycle of life and death, till it realises the Parmeshwara. And after realising Brahman the Jiva becomes Brahman itself and gets moksha.
For more, you can see this answer of mine.
- The Chariot.:
ओङ्कारं रथमारुह्य विष्णुं कृत्वाथ सारथिम् । ब्रह्मलोकपदान्वेषी रुद्राराधनतत्परः॥ तावद्रथेन गन्तव्यं यावद्रथपथि स्थितः । स्थित्वा रथपथस्थानं रथमुत्सृज्य गच्छति॥२-३॥
Mounted on the chariot in the form of Omkara and making Lord Vishnu as his charioteer (guru), the wise man, thinking of the supreme position of Brahmaloka (City of Brahman), one should always remain engrossed in the worship of Lord Rudra, the God of gods and should go by that chariot in the form of Pranava until the path that can be traversed by the chariot is completed. When that path (goal) is completed, then leaving that chariot man automatically reaches the place of the owner of the chariot (Brahman). ॥ 2-3 ॥
Here Pranava Roopi Chariot means Dharma (Righteousness).
BrahmaLoka- Abode/City of Brahman/Liberation.
Owner of Chariot- Brahman.
So, The verse means, By Making Lord Vishnu as the Guru and and by always following Dharma and keeping Moksha as his goal, One should always worship Bhagawan Rudra his whole life till his life is over and then after leaving his life he reaches Brahman itself viz gets moksha.
For more you can see this answer of mine.
Conclusion.: There is no difference between these 2 verses. The 1st "2 birds verse" describes whole life cycle of a jiva till it realises brahman and gets moksha and the 2nd "chariot verse" describes the process of how a jiva should realise brahman for getting moksha.
I hope this clarifies all your queries. Prd..