What are the weaknesses of Shamir's Secret Sharing?
let's say have used a 64-bit secret key to encrypt a file and then we split the key into 2 pieces of 32-bit


That right there is your misunderstanding. Shamir secret sharing does not split a secret into pieces that are smaller than the original secret. Instead, it splits the secret into pieces that are the same size as the original secret.

A simple graphical example typically does the best job describing Shamir's method. Assume that my secret is the number $42$ and that I want to split this into $3$ pieces to give to my friends such that any $2$ of them can get together and recreate my secret. So my threshold is $2$, so I pick a random polynomial of degree $1$ (one less than the threshold) and set the zero term to my secret. So let's say my polynomial is $f(x)=42+17x$. To generate my shares, I evaluate the polynomial at the points $1,2,3$ and get $(1,59),(2,76),(3,93)$.

I then give one share to each friend.

Now, to understand the security of this, you need to think back to the line that we used to generate these shares. If I give you 1 point on the line, what does that one point tell you about the line that was used to share the secret?
 How many lines can you draw through that one point? The answer is infinitely many. So, by having that one share, you have gained no useful information about the secret.

That is the beauty of Shamir's method. Having the less than the threshold number of shares gives you no useful information. It is only when you have that one share that you have any useful information at all.