Paleoanthropologists have long thought that there was no evidence of a major migration out of Africa at 200,000BP, so I'm not sure where that date is coming from. The first fully modern H. sapiens (FMHS) appears near Omo, Africa, dated to approximately 200,000 (Source Jurmain et al. Introduction to Physical Anthropology 2013). There were already very human-like (but not precisely human) people living in major parts of the Old World (but not Australia).

It probably took a long time for the older models to die out. For example. H. erectus survived until about 28,000BP in and around Java (same source).

The main theory (lacking no evidence for anything other than a long dwindling off of the earlier species, of which there are at least 6-7) is differential reproduction. It doesn't take a lot of extra reproduction for FMHS to slowly replace its cousins.

In Europe, where the story is best known/most data exists, FMHS arrives around 45,000BP, avoids H. neandertalensis territory for awhile (maybe as long as 10,000 years, hanging on the fringe of the other's territory) then slowly moves into Neanderthal homelands, and they begin to dwindle.

The form of FMHS that goes into Europe is often called Cro Magnon and you can look into the various reasons why anthropologists think Cro Magnon was successful (there's quite a bit of data about the comparative lifestyles of FMHS and HN. Today, most people call Cro Magnon EEMS (European Early Modern sapiens) if you want to look up more about them.

Brian Fagan is a good resource on the various theories, but they all boil down to evidence about resource use. Every place (including the Middle East) where we see HS and HN living near each other, they use their environments rather differently. Our brains are structured differently too. By 45,000, EEMS is arriving in Europe. There is archaeological evidence of their route, early skeletons are found in various places, Italy probably has the oldest but that's not where they entered Europe of course. Once EEMS appears, Neanderthal seems to vacate some of its older habitats and be confined to places that were more marginal.

Don't forget that many of the EEMS also died out in their various places, hanging on through several cold snaps, but if one looks at a particular site or sites, they are often deserted for thousands of years at a time. Human habitation in various parts of the world was tenuous (and the coping skills of EEMS were fairly sophisticated, but they were still not great in numbers).

Brian Fagan's 1996 Oxford Companion to Archaeology has a lot of the basics. Brian Sykes's Seven Daughters of Eve is a good introduction to the mtDNA evidence for EEMS for the lay person. Jared Diamond has his own well-written text that takes us up to recent discoveries.

The most recent set of "controversies" has had to do with the amount and place of inbreeding between EEMS and HN. It appears that HN did not entirely die out, some of them bred with the ancestors of modern Europeans (and Asians). Nearly everyone whose ancestors have recent (since 50,000BP) migration from Africa has some Neanderthal DNA, with the highest percentage of course being where HN used to live.

There are markers, such as the one for betaglobin, which show that we also interbred with H. erectus (so not a separate species?) See Rebecca Stefoff's book, Modern Humans, 2010, p. 49ff. (also written for non-geneticists, which is nice). Otherwise, a more comprehensive route through the fairly complex data would be Patricia Willoughby's The Evolution of Modern Humans in Africa, which gives the various data (genetic and paleontological) for the early migrations out of Africa. I assume everyone has watched Spencer Wells's documentary on youtube, which covers the main outline of the story really quickly but ignores what happened to the other hominids (Neanderthal, Erectus, Floresiensis).

The putative species, H. heidelburgensis is worth reading about as well. I say putative because its remains are not found in clusters, as if it lived in its own communities, but it arises among both H. ergaster and H. antecessor and appears able to mate with both (Jurmain et al., again).

I feel this would be better answered in the /r/AskAnthropology/ subreddit.

This isn't really an "ask historian" question so to speak.

But technically speaking, they didn't all die. We absorbed many of them into our expanding population. Everyone outside of Africa has a bit of Neanderthal DNA (a few percent). Melanesian people can trace ~8% of their ancestry back to another subspecies similar to Denisovans, who themselves had mixed with an older human lineage (perhaps a late version of homo erectus). There are hints of other hybridization events in Africa but the environment there is not really conducive to preserving DNA, which makes it harder to tell.

This figure is a bit out of date but gives this basics of the most significant admixture events. Here is the paper where that came from.

Here is a paper that suggests another two admixture events may have occured in Eurasia.