Are there advantages between different hydrocodone cough medicines?

Yes, but... Opiate medicines such as hydrocodone come as proprietary preparations, some elixers, some tablets. For a cough, and it has been shown many times that opiates work on the central nervous system to reduce coughing reflex, are sometimes mixed with Acetaminophen (linked with liver toxicity) and sometimes with other "cold" remedies. These are seriously addictive meds and should be used very sparingly.
Maybe. None are particularly effective. Recent tudies of cough medications have shown that codeine-containing cough medications work about as well as sugar pills (placebo) and other studies show hydrocodone equal to Codeine . Theoretically Hycodan (with an anticholinergic) should work better for productive coughs and tussionex (with an antihistamine) should work better if there's an allergy component.