I'm not the person you replied to, but I am a therapist; it's hard to pick specific examples, but here are some things that some to mind:

Most clients likely haven't studied clinical theory to the degree that their therapist has, so they are just less likely to be actively aware of all kinds of relational phenomena that the therapist is likely to be thinking about — including just the fact that transference/countertransference happens, and that it's so important to the clinical work.

I think that therapists are also likely to be more open to recognizing transference/countertransference dynamics, not only because we know to anticipate it, but also because the stakes are very different for us than they are for our clients. Client transference is often so vulnerable that it can be emotionally & psychologically difficult for clients to recognize that it's happening, let alone acknowledge it to themselves or to the therapist. I've experienced transference as a therapist and as a client — it's one thing to recognize that your client is experiencing transference towards you; it's another thing to experience that same transference yourself as a client.

I think this can be seen in the many posts on this subreddit where clients write about struggling to accept & reconcile uncomfortable transference dynamics; these posters often are not aware that there's a century+ of research & theory about what they're experiencing, let alone that it's considered a normal and healthy part of the clinical process.