Search itself is more functional, offering up a full list of results organized by date for a given search term, without grouping those results by person. For example, if you search for "cats" and had multiple cats conversations with someone, each incident with the cats search term is listed individually in iOS 13, which is not the case in earlier versions of iOS.
At last! Almost any messaging app has this ability to provide many results for one person you're messaging. I couldn't understand why Apple's Messages couldn't do it. It was completely unhelpful.
And yes, the new ability to reply to a specific message (as with Messenger), that would be great!