ZiIIPs (z Integrated Information Processor) and zAAPs (z Application Assist Processor) are a wee bit more interesting than that. They (and other speciality processors) are a device to prevent a chicken and egg situation created by the pricing strategies used by IBM and other mainframe software vendors, which is typically based on installed processing capacity. They (IBM et al) want to sell you software, but that software, especially if it does something very useful, will use up processing power. Bearing in mind that a competently-run mainframe will be trundling along at pretty much close to 100% capacity 24/7, adding new software will potentially require a customer to increase the processing capacity of their machine(s), which in turn increases the licensing costs for the software installed, which... well, you get the picture. Not the easiest sell.
Enter the zAAP. This essentially runs JVM-based workloads (not noted for their processor-friendliness), costs less to install/enable than a general-purpose CPU, but more importantly, does not count towards the installed processing capacity and thus incur increased licensing costs. This was obviously considered a good solution to the chicken-egg thing, because IBM then introduced the zIIP, which can (mostly) be considered a proper general-purpose processor, but the basic principle is that zIIPs and zAAPs are a cheap (in context) way to relieve capacity constraints without incurring extra software licensing costs.
Contrary to the previous poster's assertion that zIIPs can only execute specific instructions, rather they can only be used for specific workloads, using software which is specifically written to take advantage of them. IBM's DB2, for example, will use a zIIP, if available, for distributed queries (i.e. from other machines), no doubt to swallow the overhead of converting EBCDIC to Unicode, and for the building of indexes. Other vendors, if they are competent, will increasingly take advantage too. There are quite a few hoops to jump through though, and for a number of other reasons it is more or less unlikely that a customer would be able to run their normal application processing on a zIIP. Neon Software offered a product some time ago which promised to do just that and got jumped on heavily by IBM for their trouble, but as a device for taking the overhead of running system software (lucrative to IBM and ISVs) out of the capacity/cost equation, they seem to becoming rather popular.