In the context of suffix meanings, "ness" often indicates a quality or an adjectival attribute, whereas "ship" indicates an abstract concept or condition. The difference can be quite subtle, but perceptible nonetheless.
Note that "ship" is usually attached onto a noun, while "ness" is usually attached onto an adjective. This is probably a more objective, clear-cut point of distinction between "ship" and "ness" as far as their use as suffixes go. E.g.:
Friendship (noun "friend" + "ship")
Relationship (noun "relation" + "ship")
Championship (noun "champion" + "ship")
Membership (noun "member" + "ship")
These are all words that denote abstract concepts, states or conditions (e.g. The state of being friends with someone). Such concepts are not necessarily attributes or qualities, which are ways of subjectively ascribing a characteristic to something or someone. In such cases, a noun would be formed by the addition of "ness" to an adjective. This is a general rule that seems to find no exception. E.g.:
Liveliness (adjective "lively" + "ness")
Awareness (adjective "aware" + "ness")
Goodness (adjective "good" + "ness")
Sadness (adjective "sad" + "ness")
Marcio has pointed out an interesting distinction between "friendship" (the state or condition of being friends with someone) and "friendliness" (the subjective quality of being friendly). The same could be said of "dictatorship" (a situation where a dictator rules over a group of people) and "dictatorialness" (the attribute of being dictatorial, that is, like or similar to a dictator).