The purpose of Tinycore is to run in RAM.
If you make an install to a harddisk (not the frugal type, a real install) you have a system like every other system, except that it is extremely stripped down (for example busybox instead of bash+tools) and comes with an extremely stripped down kernel that may be missing some modules for your hardware to use it to its fully extent.
But if you use it in RAM, as it is intended, and add the kernel modules you need for your hardware (and of course the module-init-tools, which are missing in the base system) you will get far higher (in terms of low-spec machines) RAM usage on Tinycore than on typical distros that can run on those machines.
Don't get me wrong, Tinycore is a fine distro that has its uses (would be nice to try a 64 bit version on a high powered machine with 16GB RAM or more). Running on low-RAM machines is not one of them.

With Tinycore I could do things with my EeePC 2g Surf that no other distro would allow, that 570mhz Celeron could even do 240p Youtube videos in Flash with Tinycore. I crammed far more usability into that 2gb SSD than anything else I tried and never came close to running out of RAM. Since I had no real graphics hardware to speak of, Xorg was just bloat, Xvesa worked much better. Of course, the Eee had far more RAM than the OP's, but with careful selection of applications, you could easily stay within 64mb.

Of course, this was a customized install, but if you want to get the most out of your hardware, any distro is going to need some customization.

Yes it is, you are quite correct about that. However, it ran well for me installed on a P3 733Mhz with 48MB RAM - considering the PC was made in 2001 it was blazingly fast, even loaded with apps. But we all know a 10 year old PC with say 64MB RAM isn't gonna do much in the end today. My choice would be a non X build, pure command line - maybe on a Slackware, Alpine Linux, Debian or Arch base (or possibly NetBSD ) I have my own custom Debian distro I call CLInix and it rarely pushes much over 50MB with music playing, web browsing and IRC etc running...

While Xorg is more heavy on RAM, it supports the acceleration features of the EeePC's graphics hardware. I just sold an EeePC 4G (4GB SSD and and a slightly faster CPU) that is totally fine with playing Youtube videos in 360P under XP. But I would recommend to use programs like Minitube for such machines anyways.

+1 for that.

There is no acceleration available on that machine that made Xorg even remotely worth the weight. The only video acceleration is MPEG2, which is useful if you want to watch a DVD, useless for anything streaming. I tried both, Xvesa ran far better than Xorg. I could do 360P smoothly, but only if I wasn't doing anything else at the time, since the screen was so small 240P looked fine and was easier on resources.

According to the Xorg wiki for Intel there is 2D acceleration, xv acceleration, DRI2 support and basic 3D support, along with features like KVM, downclocking and support for the VGA output on that machine. So I would think that there is acceleration that makes at least me think about which X-server is better for that machine. If you don't have use for those types of acceleration then it is of course OK to use Xvesa. I personally tend to use Xorg on such machines, at least for the 2D acceleration.

The only things I missed with Xvesa on that system were fine control over the touchpad and VGA output, neither one was a show stopper. When you're dealing with something under 1ghz, you really need to think about every process you have running, whether the advantages it gives you are worth the CPU cycles. Xorg will take more CPU cycles than Xvesa, so what you gain in the minimal acceleration you might get from Xorg, you lose in the overall performance of the system. I did back to back comparisons on that Eee, Xvesa is faster under normal use in that case, considering the age of the OP's laptop, I would think the effect would be the same.

I ran into a similar situation with ad blocking extensions in browsers. You would think that keeping an ad from loading would help performance, but all of those CPU cycles needed to run every single element of the page through the filter means performance can be worse on a slow system. I ended up adding a Flash blocking extension which took care of the worst offenders, but didn't hurt overall performance.