"Get around to..." means you eventually do the task, but other things need doing first. "Get down to..." means you focus your attention and make the task a priority. It can also mean a point in your progress. They're really quite different!

I think you meant 'to get around to'. Short answer: they can mean the same thing. But the latter has a few more applications. Basically, 'to get around to [something]' basically means to find or make time to do something. 'To get down to [something]' can mean the same thing (to start work on something). But it has a few more applications in different contexts. It can also mean to really delve deep into something, to pay very careful attention to what you're doing. It is most often used in the following expression: 'when you get down to it', i.e. when you take a good, hard look at something.