Speciation
Speciation is a process, a process that can take millions of years. Moreover, two populations need not completely separate if the existing barriers to reproduction are "good enough" to prevent or hinder gene flow.
Barriers to reproduction can be external (separate habitats, living in different geographical areas, behavioural differences, different mating rituals, different mating seasons, mating times...) or the barriers can be internal (sperm cannot penetrate the egg, if sperm can penetrate egg the genetic differences are too big to overcome, the hybrid is unable come to term, if the hybrid is born it is sterile, if the hybrid is born fertile it is of poor health compared to non-hybrids....).
For example: Lions and Tigers can be forced to mate in captivity and produce "viable" hybrids. I say "viable" because while some of the hybrids are fertile they are of poor genetic quality and ill health. They would not be able to compete with non-hybrids in the wild. Lions and tigers also live in different habitats, and have very different social structures. These differences are good enough to prevent gene flow, they don't need to have additional internal reproductive barriers like the sperm being unable to penetrate the egg. For this reason tigers and lions are considered separate species.
Sometimes two species might look very similar, like the snipes, but con't interbreed because of one or many different barriers to gene flow. In the case of the snipes, the common snipe is restricted to the Americas where as the pin-tailed snipe is restricted to Asia. They do not have overlapping ranges, so these individuals will never meet in the wild and will never interbreed. They may interbreed say if a human brought them together, BUT thats still no guarantee. They likely have different mating seasons, mating rituals, different calls - all which would make an individual more likely to choose their own species over another similar looking species. So while these two snipe species may be able to produce viable offspring, they won't because they don't live anywhere near each other. In there case the geographical barrier is "good enough" to prevent gene flow and wouldn't necessitate that they evolve other barriers.
Neanderthals and Humans
The media really likes to over emphasize the interbreeding with Neanderthals.
Humans and Neanderthals are separate species. We evolved in two different locations (Neanderthals in Europe c. 350,000 years ago; Humans in Africa c. 200,000 years ago). We behaved differently (Neanderthals had stagnant tool cultures, Humans were much more innovative). By the time the two populations encountered each other about 50,000 years ago many external reproductive barriers were already in place.
The 1-5% DNA interchange can be explained by a couple on interbreeding events. It does not mean that this phenomenon was ubiquitous across the Neanderthal population, nor did it have to be a common event. If hybridization was not common, then this would be a good argument for separate species classification.
The 1-5% DNA has not been well explored as to what it actually does.
We do not know the context of these interbreeding events; were they consentual? rape? Moreover, we do not know how the hybrids were treated; were they accepted into human societies? were they outcasts? These social and behaviour factors can be external barriers to reproduction, in the same way that lions are social and tigers are solitary. If humans and neanderthals behaved differently, and acted like different groups (e.g. they could distinguish themselves from each other) then this would be another good argument that these two populations were well on the way to full speciation.
We do not know the vigour of the hybrids; were they all fertile? were some or the majority sterile? how fit were they in terms of being able to compete against other humans? This is important for understanding speciation.
We have no evidence that Neanderthals have human DNA - e.g. the flow of DNA appears to be one direction. This is another good indication that hybrids were of poor quality, and that speciation was well on its way to completion. "While modern humans share some nuclear DNA with the extinct Neanderthals, the two species do not share any mitochondrial DNA, which in primates is always maternally transmitted. This observation has prompted the hypothesis that whereas female humans interbreeding with male Neanderthals were able to generate fertile offspring, the progeny of female Neanderthals who mated with male humans were either rare, absent or sterile."
We do know that Neanderthal populations were already in decline in much of Europe before humans even arrived, because they were not adapting to the climate change experienced there. We do know that a good majority of Neanderthals NEVER encountered humans, and went extinct on their own accord. We do know that humans were competing for the same resources as Neanderthals, where the populations encountered one another in Southern Europe. The dominant and most supported hypothesis for the extinction of Neanderthals is NOT their admixing with human populations, but rather we outcompeted them - through passive or coercive means.
TL;DR The current evidence suggests that while humans and neanderthals were certainly capable of interbreeding and producing viable, fertile offspring, these two populations were well down the path of speciation. Several biological and behavioural reproductive barriers had already manifested by the time these two populations first met. For these reasons, and others outlined above, humans and Neanderthals are by and large considered separated species.