CAVEAT: Like most words we use to try and describe fictional universes in a more "academic" way, the terms "alternate universe" and "continuity" are not formally defined anywhere. The term continuity, in particular, can be a bit vague its possible to quibble of what is and isn't part of a given "continuity". It also has multiple related meanings (it's both an adjective and a noun.) There are some "standard" or "typical" ways in which those terms are used, but fans aren't always very precise and accurate in their terminology, particular with closely related concepts like this.
The two terms describe two different (but often related) aspects of long, complex, ongoing fantasy settings. The short answer is: a "universe" is a physical place within a setting that has a particular set of physical laws and historical events. A "continuity" is an abstract concept that encapsulates all the elements of a series that "make sense" when put together with one another.
Sometimes an alternate universe is an alternate continuity, like with the DC Elseworlds or Marvel What If... series. But other times, a continuity includes multiple universes, like the Arrowverse (where all of the alternate universes we've seen are in continuity with each other). And sometimes a single universe may have multiple continuities (e.g. the X-Men Cinematic Universe arguably has at least two distinct continuities).
So, it's not really a question of which is better, or what the advantages of one or the other are. The terms serve different purposes, and you use whichever one makes the most sense.
A "Universe" is a concept used both in- and out-of-universe (that is, by both fans and writers, but also by the characters in the story). In the DC and Marvel settings, their stories take place in a multiverse. That is, a collection of parallel or alternate universes that formed around the same time and followed similar progressions, but with varying degrees of difference. Typically these are described in very Earth-centric terms like Earth-1, Earth-5, Prime Earth, New Earth (for DC) or Earth-616, Earth-1610, Earth-19999 (for Marvel)
In this sense, a "universe":
- Is a complete, self-contained existence within the multiverse
- Has it's own complete (if possibly complicated) historical timeline
- Typically contains its own "versions" of many of the people and places from other universes
- Has a continuing, ongoing existence
- Might be potentially traveled to/from by anyone with the right technical or magical skill
"Continuity", on the other hand, is almost entirely an out-of-universe concept. It simply means the consistent set of facts and historical events that describes the behavior of a given setting. It's very closely related to canon: canon are all the things that definitely happened or could happen, according to the author(s). Continuity (as an adjective) is the degree to which canon is self-consistent, or the degree to which individual installments of a series stick to the canon from previous installments. As a noun, we also use "continuity" to describe all of the parts of a series that maintain continuity/canon with each other.
In that sense, a "continuity":
- Is an aspect of multiple parts/issues/episodes/installments taken collectively
- Could span multiple universes
- "Begins" and "ends" based on the events of a given part of a series
- Cannot be "moved to/from" because they aren't a physical place.
In some cases, alternate universe make alternate continuities, if those universes never "cross over" with others. For example, for a very long time, Earth-616 and Earth-1610 (the Marvel primary and Ultimates universes) were completely separate continuities. But those universes have interacted with each other before, so now they shared continuity: on Earth-616, it's "in continuity" to talk about Earth-1610 and the people who live there, because from the Earth-616 standpoint, Earth-1610 now definitely exists.