The defining feature of a mainframe is summed up in the word availability. This encompasses uptime, capacity for multitasking, capacity to serve many users at once, ability to service multiple large I/O requests at once, backwards-compatibility, and hardware and software reliability.
In fact, just about the only thing it doesn’t stress is raw computational power. If you need a lot of flops or a lot of parallel processing, you're looking for either a supercomputer or a cluster of some kind. A mainframe would just slow you down.
A mainframe does what it does using many technologies, some of which are being reinvented in desktop computers decades after their introduction among mainframes. The most interesting of these reinventions is virtualization, the process of running multiple guest OSes at the same time under a virtual machine. IBM was doing this decades ago with VM/370 and VM/CMS (that is, running the Conversational Monitor System as a guest under VM, their imaginatively-named virtual machine). This serves two major goals: It makes OS crashes easily survivable and debugable, and it allows very old OSes to run unmodified on modern hardware invented decades after they essentially stopped changing. (Only the virtual machine needs to know much about the hardware, because everything else has to go through it anyway.)
There have been more subtle reinventions over the decades. Web forms could be seen as a return to the block mode style of terminal interaction (as seen in the 3270 and the 5250), as opposed to the character-oriented style used in every non-IBM context to have terminals at all (as epitomized by the VT100 and its uncountable clones). The block mode style allows one server to control many more clients because all I/O is batched to be sent a screen at a time instead of being sent one line or even one key press at a time. Even with AJAX, the fundamental I/O model of Web sites looks more like block mode than anything else.
The lesson here is that the advance of technology doesn't shove the obsolete technologies completely out of the way, it relegates them to where they actually make sense. Technological advance is all about having options, which includes the option to use the “obsolete”.