Using an inductor as a LF antenna
What makes a ferrite antenna different from an inductor ?

Antennas work by electrically coupling to free space, to either radiate or receive energy -- it's their whole purpose.  An ideal inductor radiates no energy -- that would defeat the purpose of an antenna.

Can I just use a same value inductor as a replacement for my antenna ? Provided SRF and DC resistance are good enough for my use case.

No.  In general you'll get an inductor closer to the ideal inductor than a purpose-built antenna (even one on a ferrite rod).  In the worst case, you'll get an inductor that's shielded or has a toroidal core and has very little leakage.

Doesn't the size of the magnetic field depend entirely on the inductance of the coil, regardless of shape or size ?

No.  The size (and more importantly the degree to which it leaks to free space) depends on the construction of the inductor.
In general, up to at least a wavelength or so, the bigger an antenna is with respect to the wavelength of the RF you're generating, the better it couples to free space.  This means that it wastes less energy on heating itself up, and radiates more to the outside world (or does a better job of picking up external signals).
For VLF antennas, unless you're doing something heroic and possibly in violation of local airspace rules, bigger is always better.
The reason that VLF antennas are built onto ferrite rods is because they need to fit into some human-scale space -- but when you do that, you end up giving up a lot in performance.