What is the difference between doxycycline HCl and doxycycline hyclate?
[...]they prefer hyclate because it crystallizes more readily than HCl salt 


The principal component is doxycycline - in both cases. The active component is a metabolite.

Doxycycline is an antibiotic and belongs to the tetracyclines (= four rings).



In order to prepare nice ions that can traverse membranes and pumps by the body easily, pharmaceutics containing amino groups or 'alkaloids' (see the red nitrogen atom in the structure), are often converted to their hydrochloride, citrate, oxalate, bromide etc. salts.

These are salts with the (protonated) pharmaceutical as the cation and chloride as the anion. Salts are water soluble.

Here, the notation for this is doxycycline$\ce{.HCl}$.

The crystallisation is done in a solvent or a mixture of solvents.

When the doxycycline$\ce{.HCl}$ precipitates (=when it forms a solid), molecules of the solvent may get embedded, trapped, or entrained in the crystal structure (but the crystals gets pulverized anyway and re-crystallized).

Note that this is not just a wet solid! There would be little or no absorption in that case.

Exactly this happens when the hyclate is formed during the crystallization of doxycycline hydrochloride in a mixture of water and ethanol.

Hyclate is just an abbreviation for hydrochloride hemiethanolate hemihydrate, which means that each molecule of doxycycline comes with 1 $\ce{.HCl\ }$  and 0.5 $\ce{.CH3CH2OH}$ and 0.5 $\ce{.H2O}$ in the crystal.

Whether it is more concentrated (= more active component in one pill) only depends on the formulation of the respective pharmaceutical companies.