There are several reasons why atoms hybridise when bonding. Let us consider the first row elements.
1) Unhybridised orbitals differ in energy. Recall that the s orbital is more stable than the p orbital, which is more stable than the d orbital, which is more stable than the f. As a result, bonding with the individual orbitals alone would cause differences in energy in each bond, thus unfavourable altering the shape of the molecule.
2) The electron shells of atoms repel each other. Not considering the effect of bonding, the hydrogen atoms in methane would prefer to be as far apart from each other as possible, in the shape of a tetrahedron. By hybridising, the molecule adopts a conformation which reduces repulsion between bonded atoms.
3) Unhybridised orbitals simply cannot bond effectively. The s orbital orbits the nucleus in the shape of a sphere, with no direction to bonded atoms; while the p orbitals have two equivalent ends which reduce the efficiency of bonding by half. However, hybridised orbitals (sp, sp2, sp3) have one end larger than the other, so that they can bond effectively.