This is a case of present perfect simple and present perfect continuous.
We use the present perfect continuous to emphasise the duration of an action which started in the past. For finished actions we use the present perfect simple.
I've been looking for my dog. (I still look for him now.)
I've looked for my dog. (I don't necessarily look for my dog any longer.)
Present perfect simple or present perfect continuous?
We use the present perfect simple, not continuous:
- if the action is finished and complete.
I've written a letter.
- if we want to say how often an action has happened.
She's broken her leg three times.
In summary he buscado a mi perro y he estado buscando a mi perro don't mean the same.