How safe is the Certified Rotax 912 versus the Continental O-200?
As far as I can find out, no one knows. There's no good data available on engine reliability from the manufacturers - although they usually don't share that information anyway - and nothing substantial that I could find from any third parties.

There is some information online, although it's mostly opinions and there's no real, consistent data to support it. This 2008 article compared LSA engine options including the Rotax and said that there were more incidents involving Rotaxes than Jabiru engines (excluding non-engine related causes), but there were also many more Rotaxes in service so the comparison isn't meaningful.

In fact, the article said that any comparison is difficult because the Rotax was (is?) primarily used in experimental aircraft:


  The great problem in identifying engine-specific problems is that in
  the majority of installations, the engines power
  experimental/amateur-built category aircraft, where differences in
  mounting, accessory placement, cooling and vibration variations can
  instill issues that vary from one installation to another


And it also made the point that there are plenty of engine-related incidents involving Continentals:


  And accidents involving Continental O-200-powered craft were as
  numerous as you'd expect for the length and depth of its presence in
  general aviation aircraft. Here again, many occurred through no fault
  of the engine. Still, numerous reports involved engine stoppages,
  heat-related power loss and other engine problems.


The article's conclusion seems to be that a Rotax engine is not (was not) significantly different from any other engine:


  To paraphrase the words of one long-time Rotax service-center
  operator, the main problems involving the engines stem from many of
  the same types of problems—poor judgment, misuse, mismanagement and
  poor maintenance—befalling all aircraft.


I found various other forum discussions about engine noise (no conclusion), fuel efficiency (Rotax is better), maintenance costs (some preference for Rotax) and so on, but nothing authoritative. I suspect that a detailed, data-based comparison simply doesn't exist; finding anything for Lycoming vs Continental is difficult, even with the much longer history and larger installed base.

In general, engine comparisons seem to be mostly based on opinion and anecdote (as described here). That doesn't mean the information is useless, but you do have to spend some time sifting and weighing the opinions to reach your own conclusions.