Assuming a civilization has the capacity to build space vessels designed to travel from one solar system to another, what is the technological difference between traveling at 50% light speed and traveling at 100% light speed (or near)?
I'd say, pretty significant. To achieve a speed of X, you need to gain a kinetic energy of mX2 and that energy, whatever your propulsion system, ultimately comes from fuel. But since you need to have the fuel with you, that's more mass that you need to have with you when you start. In the end, it's a matter of energy density.
Then, relativistic speeds offer two important challenges that your technology must overcome:
anything in space - dust, grit, stray protons, gas molecules, junk - in your trajectory becomes a projectile hitting at relativistic speeds. You need to be able to either locate such obstacles far enough, and maybe manoeuver fast enough, to avoid them, or survive the smaller impacts.
at relativistic speeds, your ship-clock time slows down. This means that you have even less time to detect obstacles, less time to react, less time to manoeuver.
At 99% c, you send out a pulse at the speed of light towards a half-kilo pebble floating one million kilometers in front of you. The pulse takes 3 seconds to reach the pebble; in those three seconds you've covered about 895,000 km and are at 105,000 km from the pebble. The pulse goes back, and you detect it when you're at less than 10,000 km from the pebble. To move a space of s = 50 meters off from your route, hold the relativistic slow-down, you have around t = 0.03 seconds. Given that $s = \frac{1}{2}at^2$, this gives $a=\frac{2s}{t^2}$ = nine thousand gravities.
So: you either have technology to survive accelerations two orders of magnitude above lethal, and detection technology capable of locating position and speed of a pebble one million kilometers away; or a detection range proportionately higher; or the capability to survive impact, and a half-kilo pebble at .99c has the same effect of a multimegaton-range fusion bomb.
And then, what kind of technological leap is required to go beyond light speed?
The impossible kind, for all that we know. It's a sort of Chinese Corridor race: every technological leap you do will halve the distance separating you from light speed. So you go from 50%c to 75%, to 87.5%, 93.75%... but you will never reach c (the Engineer's response in the joke is "Yeah, mate, but I only need to get close enough).
So traveling 4 light years takes 4 years at light speed.
Welllll... actually, 4 light years at light speed takes no time at all, if you're aboard the ship. Time contraction again. That might be an advantage.
Of course, reaching near enough the speed of light takes time.
Would it make sense to say "We are only advanced enough to go x% of light speed"?
Yes, it makes a lot of sense.
would anything really stop anyone going near light speed given enough fuel and distance to reach that speed?
At a certain point, exotic effects become observable and begin kicking in. The most relevant is probably the Doppler-Zatsepin effect, whereby you observe the ubiquitous microwave background blue-shifted towards higher energetic levels. In other words, wherever you look you see a gamma-ray laser firing at you point-blank with energy enough to photodisintegrate the ship. This phenomenon limits the distance traveled by a fast-enough particle to what is called the GZK limit. Accelerating further will expose you to a different but equally nasty effect: the temperature of the vacuum will appear to increase.
So, relativistic travel is hot, but wearing :-)