Lapis didn't know just how important that tape recorder was for Peridot. Lapis calling it garbage was very harsh, but lessened by the fact that she didn't mean it as an ill will towards Peridot's development, but rather that such an object had no meaning for her since she just wanted to be left alone rather than prattle on and on like what Peridot loves doing.
Peridot presumably does not know about Lapis's past aside from Malachite and being familiar with Earth during the time of the CG war. Hence why her remark about "being trapped in a tape recorder too" was amusing and very poignant for Lapis as well.
Basically the meat of the episode was that both gems were in the wrong for refusing to communicate in a productive and empathetic manner. Sure, you could say that Peridot was the one who was actively trying to get closer to her, but it was done in such a way that clearly showed Peridot being insensitive about Lapis's feelings. In the end, Peridot finally got through to her and left them as per Lapis's instructions, which we can see after Steven guilt-tripping her that Lapis was regretting her attitude towards Lapis. In return, Lapis showed in her way that she too cared about Peridot by saving them from the impending HW threat, even looking back at Peridot to make sure she knows her intention.
Soooo basically Barn Mates was a fine episode at setting up a baseline for the two to reconcile and I hope the same will be done for Peridot-Jasper and Lapis-Jasper in their own episodes.
Hell, I can only hope that there will be as many episodes as needed for every gem to get their relationships explored with everyone else.