Category vs taxonomy and book
The categoryi module is very similar to the modules upon which it's based, namely book and taxonomy. This leads to a great deal of confusion over just what the difference is between category, and taxonomy and book. This page aims to explain what the differences are between these modules, and why the category module is better suited to a number of common tasks.
Firstly, let's go over what the taxonomy and book modules do. The taxonomy module allows you to define a number of vocabularies, each of which holds a set of terms (which can be arranged in a tree-likei hierarchyi). Nodes on your site can then be tagged with these terms (making them assigned nodeis). Vocabularies and terms are not themselves nodes. The book module lets you structure a group of nodes on your site into a strict tree hierarchy. It provides a contents block and other navigational elements, to help your users browse the tree of nodes.
The category module, at the most basic level, does both of these things at once. You can create a hierarchy of nodes (as a strict tree or in a tree-like fashion), and you can use those same nodes to tag other nodes on your site. It sounds simple when you say it like that. And that's because it is simple. However, while this merging of features may seem trivial, the results are not. It gives vastly improved power and flexibility for Drupal site architects.
Dynamic, categorized content is now able to have its position within the site hierarchy controlled and made predictable. Categories, categorized content, and static information pages are all able to be structured together into a single hierarchy, and this can all be done through a single uniform interface. Additional tools allow pages controlled by other modules to also be slotted into this hierarchy, without disrupting the structural rules put in place by these external modules.
The category module also has a number of features that its predecessors do not. With category_menu, a menu hierarchy is maintained that corresponds to your site's category hierarchy. Because category_menu saves all menu items to the database, you can manually change the parenti of any node, using the menu module. You can also use the menu module to bring in other pages, such as those generated by external modules, and include them in your category hierarchy. With category_display, you can control all of the navigational features that are simply forced to display with the book and taxonomy modules. You can have a table of contents show for some categories on your site, but not for others.
If you are happy with your dynamic content being tagged but not actually structured, then the category module is probably not for you. The same can be said if you're happy with the amount of control you have over a hierarchy of static pages using the book module. But if you need the features of both taxonomy and book working together, and you need more control over the whole process - and you also need full compatibility with the old modules - then the category module is your new best friend.
Taxonomy_access
I recommend that you continue to use taxonomy access. As you can see, it does work with categoryi (at least, it does for the purpose that you have in mind). Category does not have its own access control solution built in, except for the cac_lite module, which is significantly less powerful than other solutions (it's designed for quite specific needs). Alternatively, you can try using other node access modules, such as organic groups or node privacy byrole.