How important is mathematics in engineering research? In EE/EC/CS departments, there are certain fields that are theory-based (like the algo or complexity groups), ones that lean towards systems implementation (OS, programming languages, etc) and others have a component of both (like networking). 

In such a field like networking, there are professors who work on hard-core math modelling (example) and people who work on implementation and protocol design (example). From what I glean from my own paper-reading experience, the natures of these papers are as different as chalk and cheese: the math papers seem to look for ways to cast the problem in a mathematical framework and try to derive their results from such a set-up, while the implementation-oriented papers conceive of some algorithm and a protocol (based solely on logical argument rather than any rigorous mathematical premise) and present the results of their software simulations.

While on paper people argue there is no divide between theory and practice, at least to me the approach towards research differs widely between faculty members in the same field and department. Now to my questions:


How important is math emphasis in an applied field like networking? Industry work is almost always simulation-based from whatever I have seen. After all, networks are there to be implemented and deployed, so why bother about probabilistic modelling? 
When there are two modes of research in a particular field, will the PhD student's approach play a role in faculty recruitment?
Is there a widespread notion of one being superior to another? I know of professors and students who widely emphasise math and pooh-pooh "S-BAA: simulation based on arbitrary algorithm" type of papers.
How important is math emphasis in an applied field like networking?


Networking is an enormous field, which runs the full spectrum from pure mathematics to actual real-world deployment, with many shades of applied mathematics, algorithm development, modeling, simulation, and experimentation in between.  There are far more than "two modes of research" in networking, as you put it.  Mathematical emphasis is fundamental at the mathematical end of the field, useless at the deployment end, and somewhere in between in between.

To put it differently, it depends on what you mean by "networking".


Why should an engineering department award a degree to a thesis where the problem has an engineering cause, but everything is simply applied math?


There is no contradiction here.  I'd estimate that somewhere between a third and half of our engineering faculty at my university can legitimately call themselves applied mathematicians.  There was even a small but serious proposal a few years ago to move our mathematics department into our college of engineering.


Will the PhD student's approach play a role in faculty recruitment?


Of course!  Hiring patterns at my own university (in both CS and ECE) suggest that the networking PhDs most likely to be hired as faculty comfortably bridge the so-called gap between theory and application, speaking to both camps in their native languages, and applying techniques from both camps to Actually Make Things Work.  (Sylvain's answer is consistent with this observation.)


Is there a widespread notion of one being superior to another?


Of course.  Networking people are people.  As in any other wide-ranging field, many experimentalists think all theoretical work is pointless symbol-pushing, and many theoreticians think that all experimental work is mindless hacking.  They're both wrong.  Some theoretical work is pointless symbol-pushing, and some experimental work is mindless hacking, which is exactly as it should be.  Neither viewpoint is "better"; they're just different.

Not chalk and cheese, but chocolate and ginger.