How can one compare these two 4-manifolds We would like to compare the following two real 4  dimensional manifolds:
1)$M$=The tangent  bundle of $S^{2}$
2)$N$= The total space of the canonical line bundle over $\mathbb{C}P^{1}\simeq S^{2}$
Are these two manifolds homeomorphic? Are they diffeomorphic?
Some remarks: They have the same  homotopy type. So there is  no  an obvious obstruction for $M$  and  $N$ to be homeomorphic. On the other hand $M$ is  parallelizable. So it is natural to ask: Is $N$ parallelizable, too?
Is the later equivalent to ask " Is $TS^{2}\oplus \ell$ a trivial real 4 dimensional bundle. Where $\ell$ is the realification of the canonical line bundle?"?

Here's another (probably unnecessarily elaborate) proof that they are not diffeomorphic using Eliashberg's amazing theorem about which open $2$-handlebodies admit Stein structures and the adjunction inequality for Stein surfaces. I am sorry if this is not really accessible for the asker.
$M$ is the 4-manifold given by gluing an open $2$-handle along the unknot with framing coefficient $2$, and $N$ is the $4$-manifold given by gluing an open $2$-handle to the unknot with framing coefficient $1$. Therefore $\bar M$ ($M$ with the opposite orientation) is the $4$-manifold given by gluing a $2$-handle with framing $-2$ to the mirror of the unknot (which is still the unknot), and $\bar N$ is given by gluing a $2$-handle with framing $-1$ to the (mirror of the) unknot.
Now the unknot $K$ has a Legendrian embedding with $tb(K)=-1$ (with the Legendrian projection the simple one with only two cusps), hence by Eliashberg's construction of Stein manifolds $\bar M$ admits a Stein structure. 
Now we need to show $N$ and $\bar N$ cannot admit a Stein structure. If we take a disk bounded by the attaching unknot in $\Bbb R^3$ union the core of the $2$-handle we get an embedded $2$-sphere $\Sigma$ with $[\Sigma]^2= \pm1$. By the adjunction inequality for Stein surfaces, we have $$[\Sigma]^2 + |\langle c_1,[\Sigma]\rangle |\leq2g(\Sigma)-2$$
In our case this is just $\pm1+|\langle c_1,[\Sigma]\rangle | \leq -2$ which is a contradiction. So neither $N$ nor $\bar N$ admit Stein structures.
