For your first scenario I can't quite see what the big benefit of session would be, though that depends on the structure of your page and the way you build this thing in the first place. Some possible solutions I'd consider:
If the edit view opened in modal box is an external page (not part of this page or put together on the fly) then you'll have to somehow let it know what it is that you want to edit.. and in that case I'd rather provide it with page ID and field ID (or name) as GET params.
If the edit view is generated and filled out on the fly (and submitted by AJAX, I'd assume) you can just grab the content from page using JS without the need to store anything in session variables.
In your second scenario I'd also consider using GET params (specify comment ID to quote from) or storing either the comment ID or even the whole quoted text (unless you really need to support storing *a lot* of data) in JS cookie. This is an example of situation where you probably don't need to make sure that quoted text is somehow protected (in most cases user can edit it upon posting anyway), so storing that data in session provides very little extra value.
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Considering sessions vs. other methods of storing run-time data in general:
Biggest reason I tend to avoid sessions is that as the amount of data stored in session grows in size, so do the memory requirements (and, in extreme cases, also the disk space requirements) of your site. At least some PHP versions load whole session data to memory when session is started, so you can imagine what having tens of kilobytes, hundreds of kilobytes or even megabytes of data (a worst case scenario, but still) does on a site with a lot of simultaneous users.
With sessions you'll also want to make sure that you're cleaning stored session files properly (which, by the way, isn't always as trivial as it may sound) and preferably clearing stored values run-time too, especially if you're storing a lot of content. GET params, on the other hand, don't consume any extra memory.. and storing stuff in JS cookies only consumes client resources.
As for why one would prefer to use sessions, biggest benefits are that a) session variables make it possible to "hide" the mechanism behind this all from the user and b) session data is much harder (practically impossible, unless you've made it possible yourself) to tamper with (it's easy to try out different values for GET params or alter data stored in JS cookies).
Ease of use is a very real benefit too, of course. Storing data in session variables is often the least painful route.
Tradeoffs, tradeoffs.. but then again, isn't that what web / software development is always about?