This answer is somewhat of a generalization of my self answer to Why did the stars throw down their spears? where I ended up analyzing most of the poem to explain the meaning of one particular, mystifying line. It's all based on my own reading and my meager knowledge of Blake's philosophy.
The most obvious deeper meaning in "The Tyger" is how, in combination with "The Lamb", it alludes to the dual nature of the Christian creator God.
Like the other poems from Songs of Experience, "The Tyger" has a paired partner from Songs of Innocence, "The Lamb". "The Lamb" talks about the creation of the gentle, innocent lamb with its soft, woolly coat and "tender voice", and at the end establishes a correspondence between the lamb and its creator:
He is called by thy name,
For he calls himself a Lamb:
He is meek & he is mild,
He became a little child:
The language used to describe the making of the lamb is simple and far more banal than the language used for the tiger:
Little Lamb who made thee
Dost thou know who made thee
Gave thee life & bid thee feed.
By the stream & o'er the mead;
Gave thee clothing of delight,
Softest clothing wooly bright;
Gave thee such a tender voice,
Making all the vales rejoice!
Essentially, "Who gave you your life? Who told you to feed? Who gave you your wool and your voice?"
Even though "The Tyger" begins with a similar passage, enumerating the features of the tiger and asking who made them, there's a certain conceit to the language that has no parallel in "The Lamb", most obvious in the poem's fourth stanza:
What the hammer? what the chain,
In what furnace was thy brain?
What the anvil? what dread grasp,
Dare its deadly terrors clasp!
The tiger is described with military and industrial language. It was constructed in furnaces, on anvils, with hammers and chains and ropes ("And what shoulder, & what art, / Could twist the sinews of thy heart?"). Unlike the lamb, the tiger is a weapon of war, adapted for killing.
But even though we've just spent four stanzas wondering what immortal being could possibly have crafted a weapon of war as fearsome as the tiger, the fifth stanza suddenly admits the possibility that it was the same being who made the gentle and innocent lamb, ending with the line "Did he who made the Lamb make thee?"
The last stanza repeats the first stanza, but this time instead of asking who "Could frame thy fearful symmetry", it asks who "Dare frame thy fearful symmetry". The narrator is implying that God, who made the Lamb, is perfectly capable of also creating the Tiger, but wonders that he dared to do it.
On a deeper level, the pair of "The Lamb" and "The Tyger" can represent a person attaining an adult, intellectual understanding of God and the nature of the world. God loves the little children, his little innocent lambs frolicking in their fields and knowing not how to do evil. To them "He is meek & he is mild". He is the God of love from the New Testament, who "calls himself a Lamb" and "became a little child". But the adult is closer to the tiger, capable of war, killing, and sin more broadly; and sometimes sinful adults meet the Old Testament God who smote Sodom and Gomorrah, the vengeful and destructive force who "smiled his work to see" when "the stars threw down their spears / And water'd Heaven with their tears".
The second stanza also offers some support for this reading:
In what distant deeps or skies.
Burnt the fire of thine eyes?
On what wings dare he aspire?
What the hand, dare seize the fire?
It sounds a lot like the myth of Prometheus, who stole fire from the gods to give to humans. Fire was the beginning of furnaces, hammers, anvils, chains, and the other industrial paraphernalia that are mentioned later, so attaining fire was the moment humans became able to create weapons of war to become like the tiger. But attaining fire in the Promethean story also has some resemblance to the fall of Adam and Eve in Christianity; it was a sudden event in which humans gained something originally reserved for gods, and became a little bit like gods themselves, "growing up" in a sense. It's a common trope in Christian thought that children live in an Edenic, sin-free state and don't fall and become sinful until they grow up, so growing up is linked via the fall of Adam and Eve to attaining fire.
In other words, God, in one of his moods as the vengeful God who punishes sinners, created the tiger, a fearsome creature of war. Humans, after the fall of Adam and Eve, gained the same capacity for vengeance and war, but doing so opened them to being smitten by the vengeful God in turn. The story of how the tiger was created represents this concretely with attaining fire as the fall of Adam and Eve and the industrial tools that fire enabled for creating weapons and wreaking destruction representing the sin that humans became capable of. By realizing that God created the fearsome tiger as well as the gentle lamb, a person can gain an adult, nuanced understanding of God by comprehending His dual nature.
