This is the thing I've wanted to really talk about for... years, but it's hard trying to figure out how to go about it, though I was able to work my way through it in casual conversation recently so let's give it a whirl in writing.
We seem to live in an age of absolutes, of increasingly polarized thinking. More than ever people argue that their ideas are right and everyone else's are wrong, wrong, wrongity-wrong! I wish I could take the knowledge I picked up from my university experience as a whole, and make everyone see it, even if they could never hope to pass the classes. Because in science, theories just don't work like this, especially the big famous ideas about how the world is put together.
A scientific theory (usually by which scientists usually mean a mathematical model) is an attempt to figure what reality is. It's really an "educated guess", and judged by whether it can make accurate predictions about the world we live in. It's rare for any good theory to be completely disproven, even when we find out later it's wrong there are many different relationships two theories can have. Some examples.
Geocentric vs Heliocentric Model of the Solar System
The "popular science" version of this story is that for centuries, everyone knew the Sun went around the Earth, and though it was dead wrong, it was entrenched as dogma by the Catholic Church until brave Galileo challenged their stifling authority.
The thing is, the Geocentric Model isn't all that bad! Using its crystalline spheres, cycles, and epicycles, one can calculate where the planets will be in the sky centuries ahead of time, predict solar and lunar eclipses, and so on. It's just that it's got the mechanism the solar system works by all wrong, and all those circles in it are imaginary, but useful in performing the calculations. The Geocentric Model didn't last so long because of dogma alone, it was invented and used because it was the first method Western Civilization had to predict things like eclipses... and in so doing dispelled a lot of fear in ancient people who had no idea when or why the Sun might stop shining, or if it would come back. Seeing that it all was clockwork, and would go on regardless of anything we did (or didn't) do here on Earth went a long way, even if people didn't yet know the real reason why the planets moved.
The Heliocentric Model that replaced it was indeed a breath of fresh air, not just philosophically but mathematically, by making the orbital calculations a lot simpler. It still didn't have the underlying mechanism for the movement of the planets. It could say what path they took but it wasn't until Newton's Theory of gravity, and all the physics since, that we understand why planets circle the Sun.
Newton vs. Einstein
Einstein's Theory of Relativity didn't replace Newton's Laws of Motion either. Far from it, it expanded them, at least mathematically. (I only had one chapter in my physics textbook on this, but it was enlightening.)
Newton's main equation, the one mechanical engineers use all the time, is F = ma, Force equals Mass times Acceleration. A good way to visualize how this works is its extremes. A bullet packs a punch because though it's a tiny mass, it's moving at high speed. A train of 40 boxcars coasting in a rail yard has a huge, dangerous force, because even though it's so slow you can barely tell its moving, it's a big mass.
Einstein has several equations, but the main one to calculate how fast an object ends up moving for how much energy is "pumped in" (this is where it would take infinite energy to reach the speed of light, so nothing with mass can do it, just massless photons) is built around F = ma. Now there's a trick in physics and engineering where, if a number is approaching a value, say zero, when it's almost zero, scientists just say, "close enough" and slash it out with an arrow pointing to that value. A low speeds, the kind we deal with in everyday life where on Earth, you can slash out most of Einstein's equation (with those diagonal arrows to zeros) and voila, the terms that remain are good old F = ma. (This is, I think, a taste of what some of the more "out there" physicists are talking about when they say the "math is so beautiful, so elegant" in a particular theory.)
This is an important part of any new scientific theory, actually, the ability to not just explain something new, but agree with what we already know to be true.
This is one of the big mysteries of our age. Are these things we're trying to understand in Quantum Mechanics waves or particles or both? I've seen talk of particles as "energy packets" with loosely-defined edges, and other interesting ideas. Sometimes they seem to act just like particles and have to be mathematically modeled as particles to accurately predict what they'll do. In another situation, they seem to act just like waves and have to be mathematically modeled as waves to accurately predict what they'll do.
And then there's the famous Double-Slit Experiment which still has scientists bouncing around ideas to try and explain it... the part that blows my mind is to think everything in the Universe, including me, is made of this stuff that acts so strange.
The point is, neither the Wave Theory or Particle Theory is right or wrong per se. One has to use one or the other for different applications of physics and technology, and it's even possible someday we'll invent a new concept, not particles, not waves, to explain what this stuff really is. (Actually String Theory is a an attempt to do just that... it just hasn't been conclusively proven if the strings are really real or not yet.)
General Relativity vs Quantum Mechanics
Right now, there are two major theories (and sets of equations) to model reality. General Relativity is the familiar Newtonian physics we see in everyday life, expanded upon by Einstein, and it works well at really big scales, modeling the motions of Galaxies in the Universe. Quantum Mechanics works well for really small scales, modeling what the particles in the Stars are doing when they shine. The problem is neither one works well in other's domain.
General Relativity would suggest electrons orbit atoms in neat tidy paths, but they seem to really be able to turn into "shells" of probability around atomic nuclei, "tunneling" to different energy levels as if they can teleport, and even sometimes be in two places at once to interact with themselves. Objects in everyday life don't behave like this at all, so Quantum Mechanics doesn't apply to well to large size scales.
(Though as an interesting exercise, my textbook used a quantum mechanics equation to calculate the wavelength of a thrown baseball as if it had wave-particle duality. It turns out to be so small it could never be detected. Whether all moving objects really have such tiny wavelengths is what scientists call an untestable prediction, and since mathematical models are judged by making useful predictions, any scientific theory that can't do that is rejected by default. That science in general is practical like this is lost on some people, fueling some of the epic internet debates I've seen.)
Every Curve is a Straight Line
Not a big theory, but this came up now and again in my engineering courses. Often, while a theory of a physical phenomenon might produce a mathematical curve (and these were usually commonplace everyday stuff, like the rate of water flowing through a pipe, or how much a block of steel bends under a certain amount of force) it's ok to pretend it's a straight line sometimes.
Mathematically, if you "zoom in" on a small piece of any curve, you can get a reasonably accurate model of the same thing with a straight line cutting across that portion. And in some engineering applications, our technology rarely deals with the extremes, so it's common practice to use simpler mathematical models. It's just taught that we need to remember in the back of our minds if we ever go to the physical extremes, the simple models won't be accurate anymore.
The little straight line isn't "wrong" and the big curve "right", not if the end result is bridges that stand up and airplanes that stay in the air.
And there you have it, folks
These are some of the relationships ideas can have with each other. Even "Wrong" ideas defeated long ago, like the Geocentric model still work, and often ideas compliment each other, instead of competing, like wave-particle duality. The two ideas, (They're Particles! No, they're waves!) taken together, get us closer to understanding the Universe we live in than either ever could alone. (And we're talking scientifically testable theories about how the world works on a practical, physical level, not even into really subjective stuff like religion and politics!)
It's something I wish all the people arguing politics and religion knew more about. Religion especially, because whether or not there's a God, and an Afterlife, and spirits and magic, are also theories about how reality works, just as much as science.
If I knew how to explain this all better, I'd make a movie out of this that everyone could understand, formal education or not.