the first one means to carry on with or carry on doing something the second means to persist, to continue in the sense of preservation if you are learning chinese some tools to help you http://www.chineselearners.net/OnlineTools/Downloads and press the big yellow button

持续 last. The rain 持续 five days. 继续 go on. Let's 继续 our conversation.

继续：stop doing one thing, then continue to do it again. 歇一歇，再继续做吧。 持续：no interval, doing one thing all the time. 雨持续下了一整天。 The both can not be used at the same time. I hope the explanation can help you.

持续 means last. 继续 often use for example" let's 继续 (go on with) our work.

we often sth 持续 ，to express sth last (time,situations and so on) eg 雨持续了好久。 the rain lasts fof a long time while 继续 1) can have the similar use(there are diffierence in customs) eg this thing go on . 事情继续发展 2) express sb continue doing sth welcome asking :P