In the past the SRi models typically had different gearing and harder suspension than other models. That's no longer the case with the Astra K so the Sport button is the only step in that direction. It apparently changes the feel of the accelerator and steering but I haven't tried it. I don't think you're missing a lot though.
Note the Elite has AGR Ergonomic passenger seat, 2 rear USB charging sockets and heated outer rear seats which are not available on any other model even as options.
See the
New Astra Price/Specification Guide 1 January 2016 for full spec details - you can download it here:
https://www.vauxhall.co.uk/tools/vauxhall-request-brochure.html
It includes a detailed spec and price for each option.
You need to read it carefully to see that the details like the Parking Pack offers electrically foldable mirrors, the front/rear parking sensors add 6 speakers to Design and Energy models and the sunroof changes the SRi roof headliner from black to light grey.
It's not obvious until you look through it carefully.