- if you're talking about a field of study or an area of knowledge, that's a domain, not a dominion (e.g. I specialise in the domain of theoretical physics). This is also probably where the computing/Internet usage of the term domain came from.
- domain is also used in magnetism and in maths - in neither case can you use dominion
- dominion (but not domain) is used when talking about having control/power/command over something (eg. to establish dominion over nature)
While both words originate from the same Latin root of dominus (a lord, master), dominion is closer related in meaning to a third similar word, to dominate, while domain developed separately, originally denoting the land or property of a lord/master. As such, domain concerns area/territory, while dominion concerns having control over something - and yes, that results in some overlap.
However, only in one case is there a possible overlap between domain and dominion, and that is when we're talking about a territory or land that is controlled by some ruler or government. When to use which in this case is beyond me, though there are particular areas/territories that are generally referred to as one or the other.
Also, note that some countries in the British Commonwealth used to be known as Dominions - eg. the Dominion of Canada. As this was a proper noun, Domain of Canada would not be an acceptable synonym here.