The phrase "sich jemandes annehmen" or "sich etwas annehmen" is relatively old-fashioned. You may want to note that it uses the genitive, so it's "sich jemandes annehmen", not "sich jemandem annehmen".
As for the difference to "sich um jemanden kümmern", in my experience it's mostly a matter of different times that are referred to. "Sich jemandes annehmen" refers more to a point in time, like the time you decided to take care of somebody. "Sich um jemanden kümmern" refers more to the actual acts of care of a span of time after the decision. Otherwise, the phrases are quite similar.
I'd read the quoted example as a sequence of events, like
Ich habe mich der Katze angenommen, und seither kümmere ich mich um sie.
I took responsibility for the cat, and since then I've been taking care of it.