Inflatables: Much more stable at rest (no rocking.) Much lighter than a similar size hard boat (applies, but not so much, when comparing to an aluminum hard boat, though.) Higher load rating than a similar size hard boat. Self-fendering (to a point.) No gel coat to constantly repair. Smaller boats can be broken down and packed away in the garage. Nearly unsinkable unless you have a high degree of boat trauma.
Hard boats: Won't get you stared at as much (in the US at least.) Easier to bolt on accessories. Not as likely to have someone with a knife do damage to it. Easier to find a boat with cabin space. Generally heavier, so marginally more comfortable in larger sea states (to a point, and depends on hull shape.)