I think one important difference is that back then they focused less on the technicalities of a work / performance and more on the overall impression it made on the audience. Read their reviews, starting with the (in)famous "The Instrumental Music of Beethoven" by ETA Hoffmann. Although he was a composer himself, he doesn't write one single word about tonic-dominant relationships, modulations, cadenzas and other technicalities which make the bulk of most of our CD booklets today; Schumann, ditto; Berlioz, ditto; they write almost exclusively in terms of thoughts, feelings, moods, images and impressions --- nay, they actually wax poetic about them (the only booklets I've read that can compare, albeit palely, are those by Evgeny Sudbin and Fazil Say --- not surprisingly, they are themselves performers, not musicologists). Back then they saw, and appreciated, first and foremost the forest, even at the price of a few trees sacrificed here and there. Today, with our obsession of finding the perfect performance in the perfect sound, the forest is almost negligible, provided all trees are green, tall and regularly aligned.
Witness to this fundamental difference: Cortot and Schnabel. Both were members of the "old guard", both were notorious for their flawed technique --- just yesterday Schnabel was labeled "incompetent" right here on GMG (I'll pass on the competence of that poster to really judge S's competence) --- yet their recordings of Chopin and Beethoven are universally acclaimed as monumental, and all those who praise them do so in terms of artistic, intellectual, spiritual and emotional terns, not in terms of how they negociate tonic-dominant relationships, modulations, cadenzas and other technicalities.
What I ultimately mean is this: comparing at leisure several performances of a given piece in order to find (oftenly minute) differences, or technical flaws, and therefore establish who "owns" it is not what they generally did back then, but it's what we generally seem to delight in doing. And this has been brought about by the advent and the perfectioning of the recording technology and this, I think, is THE most important difference between us and them.