Wrong. Where do you keep getting this from?
The two are not anywhere near the same, as easily evidenced by the fact that we can build suspension bridges but no materials we have right now can build a space elevator.
What we can build right now is irrelevant. In terms of physics and construction technique, it's a suspension bridge -- a rather unique one, because the cable itself is the bridge, but still a suspension bridge. Instead of towers, it has a planet and (possibly) an asteroid as the points it hangs from.
The only similarity they have is that gravity aids in the tensioning of the cables. The cables themselves are entirely different.
And that's the point! A suspension bridge made of jungle vines, of cotton clothesline, of hemp rope, of steel cable -- they're all the same thing. The materials don't matter, the height doesn't matter, the length doesn't matter -- only the equations matter. The important items come down to self-length, tensile strength, cross section vs. distance from base, which are just plugged into the equations.
They're the very same things the engineers on the Golden Gate faced: you write your equation for you span, you plug in your values for your material, and you know if your bridge will hold; conversely, you design your bridge, plug in known values, and get a figure for how strong a material must be to do the job. It's no different than square buildings sitting on the ground: the equations are the same for a beam-and-post hall in the fourteenth century and a brick and concrete fourteen-floor structure of the last century -- or the concrete-and-steel ones of today.
Yet the cables for a space elevator will be very, very much like old hemp ropes: fibers bound by friction and possibly some matrix are spun into and hold together as thread; threads are braided into cord; cord is stretched out to cover the span, and then more cords are twined about them to make a cable. That's the way suspension bridges are done -- a lead line, twined about to be thicker and thicker, until the desired diameter is achieved (or occasionally, a lead line will serve as carrier for the main cable to be pulled across the span).
Are you kidding me?
Lifting it is one of the primary problems facing the construction of a space elevator. None of our current launchers can lift as much mass as would be necessary to lift it. So you would have to bootstrap it with a smaller one, which presents a whole other set of construction design issues.
Huh?
Oh -- I said "lift", not "lifting" -- look at the context: turbulence, drag, lift -- all nouns having to do with aerodynamics.
As for bootstrapping -- exactly. The lead line would be only large enough to have a long enough self-length and carry the twining machine. The typical proposal is to drop the lead line, catch it (trade-off: make the ballast mass too big, the spin unit at the top has to do more work and be correspondingly more massive itself; make it too small, and it could be dangerous to catch), then drop the first twiner from above, braiding three identical cables around the first, followed by an ascending twiner braiding yet more, etc. Note that this requires a launch payload more than four times the size of one merely dropping the lead line -- but if the spinner only drops one line and a twiner is sent up, the lead line has to have a different design.
The only technical problem there is twiners that work in vacuum -- twiners to work with all sorts of cable materials already exist.
BTW: for fun I went looking but couldn't find anything regarding an idea I read about a while back--
Some researchers have made synthetic spiderweb that's got a self-length in the hundreds of kilometers [self-length is the length of the longest piece of a given material than will stay together in one gravity and not break under its own weight ~ as examples, the self-length of our best steel cable is around fifty kilometers and of a Kevlar one around four hundred kilometers], with a side benefit of sticking to whatever it contacts. It's been suggested that using fibers of the spiderweb as a matrix for nanotube fiber would give the cohesion which has been lacking in many nanotube experiments. Development of nanotubes with protrusions may make that unnecessary, but it's a fun idea.