The line is becoming increasingly blurred as more and more traditional 1570 IMAX film venues are converting to digital, maintaining the standard IMAX screen-size, but now with either a digital or laser projector. The term "Lie-MAX" came about because when IMAX first made a deal with prominent theater chains like AMC and Regal to install "IMAX" screens and sound systems in their regular theaters, most of those screens were only 1/4 the size of a proper IMAX screen, and didn't use 70mm film like the real ones. THOSE are what we used to designate as digital IMAXes, or LieMAXes. And they are still probably the most common ones and far inferior to the "real" IMAX experience, imo. However, as I said, a lot of the "real" IMAXes are now making the inevitable conversion to either digital or laser projection, so the difference in image quality with LieMAXes isn't as apparent as it used to be - just a (massive) difference in screen size remains.
Here's an article discussing the difference:
http://www.ifc.com/2011/12/imax-guide