Strangulation (done (in)correctly) can cut off the blood flow to the brain, which halts oxygen delivery to the tissues. Holding your breath just causes oxygen delivery to the blood to stop. Your body has enough blood and oxygen in it to last for a few minutes without breathing, but only as long as it is circulating to the tissues that require it.

While holding your breath, the ratio of carbon dioxide to oxygen will steadily increase, which will eventually trigger some chemoreceptors and force a reflexive diaphragmatic contraction.