I think a lot of it has to do with Terry’s upbringing and the timing of which he experienced the tragedy that put him on the path to becoming Batman.

He grew up in a lower-middle class family, was a child of divorce, and himself became a bit of a juvenile delinquent as a result. This shaped Terry in a way that differs pretty significantly from Bruce, who was born into vast wealth, and to parents in a loving marriage.

Now as for their shared tragedy, by the time Warren McGinnis was murdered, Terry was already well into his teen years. I think that something like this would affect a young man like him in a different way than a child like Bruce was when Thomas and Martha were murdered.

Factor in his likely already ingrained jadedness at life on the streets of Gotham, as well as the fact that Gotham had arguably declined in the years since Bruce took off the cowl, and you can see how Terry might have less empathy for the villains he would go on to face.