The obvious difference between a libertarian government and divine monarchy is in the level of authority given to the government. In a monarchy, the monarch's will is law. Under libertarian government, the government is extremely limited to only protecting a very small set of rights. You could in theory have a libertarian monarchy where a king was only given a limited authority to enforce laws related to natural rights violations, but the two concepts are very different.

A libertarian democracy of course is more desirable. Which means the people's votes, or the authority given to their representatives, would be limited to only those issues related to natural rights violations. Under a libertarian government, the people are still quite free to exercise their will, only they could not use the government to impose their will on other individuals. Social problems can still be solved through community organizations outside of the government.

I think the biggest weakness of the pure libertarian philosophy is the uncompromising defense of property rights. If a small group manages to acquire all of the property, they can use their monopoly to exploit everyone else while the government protects their "rights".