I fall right in the middle of this disagreement. Here's some notes, and an idea.

There are good reasons to have the search-option-preferences in both locations (eg. easier-discoverability for newcomers, familiarity for oldtimers), 
and there are good reasons to centralize/unduplicate at a single location (with Special:Search->Advanced being the reasonable contender because the options-list *has* to appear there).

We do need easy-discoverability for newcomers. 
Searching for help documentation is HELL unless we add the Wikipedia: , Help: , and Template: namespaces to our default search. For the first few years as a new editor, I was constantly searching those namespaces for tools and guides. (Note that the Template: and Category: namespaces aren't included in the "Help and Project pages" output)

I'm surprised that we don't already have a link to [[Special:Preferences#mw-prefsection-searchoptions]] anywhere at [[Special:Search]] (advanced tab or otherwise)! How did we miss that, for so long?!  Forest, trees.

I do also prefer the simple process for "Save" at Special:Preferences. It's not perfect (see {T57966}), but it's fairly intuitive. 
The new system isn't intuitive. (It makes sense once explained, but an explanation is needed...)
I think the problem with the new version, is the physical distance, or dis-connection, of the checkbox from the search button... 


So, 2 fuzzy/rough suggestions:

A) Add a new Blue "Save selection as my default, and Search" button, at the advanced-search tab, instead of the new "Remember selection for future searches" checkbox line. 

B) leave the "Search" tab at Special:Preferences, but 
B.1) just link to Special:Search->Advanced from within it.   
B.2) Or, when clicked, have it open Special:Search->Advanced in a new tab/window.   (New tab, because the user might have other unsaved preference-changes, so we can't change the current page. (cf. {T57966}))  
Either of these would solve the issues of discoverability and change-aversion.  Yes, it'll still leave an annoyingly almost-empty tab, but it's *way* better than the current/old layout, ie. comment #0

--

All 4 links, for your quick comparison:
https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-searchoptions
https://en.wikipedia.org/w/index.php?title=Special:Search&search=&fulltext=Search&profile=advanced
http://en.wikipedia.beta.wmflabs.org/wiki/Special:Preferences
http://en.wikipedia.beta.wmflabs.org/w/index.php?title=Special:Search&search=&fulltext=Search&profile=advanced