Those are almost the same in meaning, but おかねがない tends to be used for relatively long term situation (in other words, in the same nuance as "poor"), whereas おかねをもっていない is for the current situation (e.g. "I forgot my wallet at home"). Of course the context is the king, so いまてもとにおかねがない "I have no money in hand" is possible, though.
Some intangible words, for instance 関心 concern, 興味 interest, 野心 ambition, 知識 knowledge or 技術 skill, can be used with ~をもっている, so it depends on the word.