I test few CMSes to choose one for my website. My provider sets only 8M to script memory usage. TikiWiki installs and works fine, but BitWeaver needs at least 16M to install. Why is that since BitWeaver seems to be more lightweight? Can I achieve installation at all?

using a 32 bit operating system, 8M is just barely enough, but should work. however, on a 64 bit operating system, memory consumption is increased signifcantly in PHP for the exact same code running.

8M is ridiculously low, especially for a 64 bit host. with 8M you cannot do any image processing of uploaded thumbnails. get your host to up you a bit, install php-eaccelerator (which reduces memory consumption by 80%), get a new host, or do all 3.