Backlight with no image, huh? Interesting. Standard TN panels are often naturally white, while IPS panels are usually naturally black. My first thought was that you weren't hitting the power pins, but (unless it's an IPS panel) a black screen would seem to indicate that it's being powered correctly.
I guess let me back up a minute. Is LVDS universal? "Sort of". Assuming you have a constant color depth and both systems use the same bit ordering, signal-wise LVDS is usually compatible between systems. There's no real "encoding" to speak of, no error correction, etc, the signal is pretty much raw - so even if the color depth and bit packing are off, you'll usually see "something", certainly not a blank screen.
What is certainly not standard is the physical interface. I have examples of more than a dozen different connector styles carrying LVDS, and inside that set I have several examples of identical connectors with vastly different pinouts. I even have an example of two otherwise identical panels, separated by only a (xx) revision suffix, with different pinouts and even completely different connectors (actually it's the LG LD070WS2 I mentioned in another recent thread). That's understandable - different OEMs may want the same type of panel, but will need the connection in a different place depending on the design of the end product - but what boggles me is when companies use the same connector in different ways on different products. For instance, Apple puts the same 30-pin I-PEX Cabline-CA connector on several of their systems but uses vastly different pinouts between them.
So, yeah. While the connector may have fit, it won't necessarily be providing the right signals to the right places. I'd venture a guess that you got lucky and the power pins were in the right places, based on the description you've given. But because the display is blank, I suspect one of a couple things. It is possible that the resolution is incorrect, as st2000 guessed, but that wouldn't be my first choice. The pixel clock rate from the resolution of a "typical" 15.6" panel shouldn't be outside the usable range of a "typical" 17" panel (unless the 15.6 is a super-high-resolution type). Even then, I'd expect the timing controller to be able to make "something" out of the signal, not just give up and shut off.
My first guess is that your clock signal is in the wrong place. No clock means the tcon will ignore any other data being presented to it. It could also be that your connectors are a pin offset from each other - as in, there's an extra pin such that pin 10 on your laptop matches the signal for pin 11 on the donor screen - which would typically result in one half of each differential pair being grounded and presumably no usable signal making it in.
Really there's a thousand different ways things could be mis-connected - the most useful thing to do would be to pull the model numbers from both panels and see if you can find datasheets for them, this will tell you the pinouts and from there you can see where things are going wrong (or post the models here and one of us can take a look as well).
Oh, and re: voltage. Yeah, a lot of times the larger panels are indeed 5V and the smaller ones 3.3V. It's no set rule, but it does make some sense: the larger panels draw more power to drive their row and column drivers (since they either have more pixels or have to twist more crystals per pixel due to increased pixel size). The wires or traces carrying signals and power to the panels are very small, so to avoid running very large numbers of wires/traces, the larger screens are delivered higher voltage which they sub-regulate onboard. But there's no rule that says "screens larger than 15 inches will be 5V" etc - it varies based on power requirements, connector type and manufacturer whim.
(extra extra aside: There is also a slim chance that one display's interface is embedded DisplayPort and the other is not, and that's why it doesn't work - because they're speaking different languages. If you have the model numbers, though, that's easy enough to rule out. It is however unlikely for them to be TTL - I will make a bold statement and say no recently-manufactured laptop drives a raw TTL display, LVDS has far too many advantages from signal integrity/EMI/cross manufacturer compatibility/etc standpoints)