The Google site detects that you are using an iPhone and redirects you to a page specifically designed for phone browsing. That is true of a lot of Google's sites.
The iPad goes to the standard Google Search site, so it behaves just like a normal desktop browser. Google does redirect iPad for some of it's sites, but when it does it does not (normally) go to a phone site, but one specifically designed for tablets.
If you really want the iPhone site(s) on your iPad there are some third party browsers than will lie to Google about what kind of device they are. Atomic (not the lite version) and iCab Mobile are two of the most popular.
Using Atomic Browser, tap on settings (cog) icon, tap on Identify Browser As, and choose Mobile Safari - iPHone. The site looks kind of weird and squashed, but it works. It looks a bit better in portrait mode than landscape.
There is a similar setting for iCab called Browser ID.