When would AXI4Lite be a better choice than the APB bus?
The substantial difference between AXI-Lite and APB is that AXI implements independent read and write addresses. So there is a little bit more control signalling, and probably a ~33% increase in routing/buffering resource. 

I would expect your existing APB peripherals use a 32 bit data bus, so the conversion to AXI would be less trivial than you anticipate - unless you have suitable AXI interfaces which can be re-used already.

Transitioning to AXI should improve bandwidth, and if you are able to transition 100% should simplify ongoing support. If the peripherals in question are low activity, or you need to retain APB in your design (for debug components or similar), then the effort of migrating looks less likely to give a good return.

Its worth noting that Cortex-M processors still use AHB, and APB is used for both debug and (much less often) configuration interfaces of some peripherals.