Apparently, "eu" is the subject in "eu gosto (de isso)" while "me" is the object in "me gusta (algo)". Why such a difference between two languages? What's the historical evolution of this expression?
Superficially subject and object markings do not always correlate with semantic "subject" and "object".
For example in English:
I won the election.
It's clearly everybody else (unless you too went to polls) who made you that way. So the grammatic subject "I" really stands for some experiencer/object first person.
In your question example, one of the lang's is more "transparent" as for this.
By the way, Icelandic has crazy constructions that mix both of these, like "me.OBJ like.SG. horses.PL" or so.