PageRank is an algorithm created by Google founder Larry Page and named after him. It’s a simple measurement of links and link equity. Google looks at a given site and measures the number of links pointing to that site. It measures the authority of those sites as well, to prevent abuse from link farms. A site with higher quality incoming links in a higher number will have a better PageRank.
Authority is a metric created by Moz.com as a more accurate indication of the value of the site. Like PageRank, it is a logarithmic calculation of the authority of the site. Rising from a low 30 to 35 will be easier on this scale than rising from a high 80 to 85. A DA of 100 is the top ideal, while an unranked site would have a DA of zero.
Domain Authority is affected by many of the same things that affect your SERPs position and your user engagement, which makes sense. DA is a measurement of your SEO, popularity, and engagement. Better metrics all-around will increase your Domain Authority.
Keep in mind, also, that Moz offers the same calculation on a page level as well. Where DA and PR only work on the site itself, going by the domain, Page Authority works on the individual page. This can be a good way to analyze the breadth of your site. Which pages have low authority and are keeping your ranking down? Which pages have high authority and what are they doing right? When you analyze your site in this light, you can see even more value in the metric.
The bottom line is that Domain Authority is more active, up to date metric that ranks a site in a more granular way. A higher DA is better than a higher PR, particularly now that Google is pulling back on its investment in PR.