Cambridge may have oversimplified the matter somewhat. The difference is in the suffixes, which both operated upon the root word obsess.
obsess To excessively preoccupy the thoughts or feelings of; to haunt the mind persistently. 1
-ive An adjective suffix signifying relating or belonging to of the nature of tending to; as, affirmative, active, conclusive, corrective, diminutive .1
-ed The termination of the past participle of regular, or weak, verbs; also, of analogous participial adjectives from nouns; as, pigmented; talented.1
Thus the word obsessive relates more narrowly to the tendency to frequently think a certain thought or feel a certain way.
The -ed suffix is a little more complicated as it acts as a past participle.
Participle (Gram) A part of speech partaking of the nature of both verb and adjective; a form of a verb, or verbal adjective, modifying a noun, but taking the adjuncts of the verb from which it is derived. In the sentences: a letter is written; being asleep he did not hear; exhausted by toil he will sleep soundly,written being, and exhausted are participles.1
Since participles have the partial nature of a verb, it indicates some sort of action which implies an agent to cause the action. This is not so different so far but participles often denote an ongoing state of action, and past participles in particular further note that this state has at the least already began, if it has not yet also subsisted too.
Hence obsessed indicates that there is obsession occurring within a certain period of time. This places emphasis on an act of obsession itself and its actual cause, rather than having the latent ability to become obsessed or potential to cause obsession like the word obsessive does.
1 Webster's Revised Unabridged Dictionary 1913