If you started law school yesterday and are already entirely lost, join the club. So is everyone else. Give yourself a few weeks of not getting it.
That said, I'll try my best to answer. As others have said, this distinction has disappeared today and the difference is pretty much irrelevant for your intents and purposes. You don't need to know all those old writs of action for exam purposes, but (as best I remember it from my 1L classes last year) trespass on the case was for indirect harms. For example, let's say I'm a logger riding along and some of my logs fall off my truck, roll onto your land, and knock down your fence. In such a case, you would've wanted to sue me in a direct trespass action. But let's say, I'm a logger going along and logs fall off my truck and land in the street. An hour later, you come along and your horse trips over the log and you're thrown to the ground. That would've had to have been a trespass on the case action. I kind of made those two examples up, but I think that's the basic idea.
Similarly, if I had a horse in my barn and you came and stole it from me, then in the old days I'd sue you in trespass for the value of my horse. But if I hired you to equip my horse with horseshoes, but you put them on wrong, and as a result several days later my horse got an infection in that foot and I couldn't use him in my business, then I'd sue you in trespass on the case for the benefit I lost.
The basic idea, as I recall, was that in the old days (long before our reformed civil procedure system abolished all forms of action except for the "civil action") you could risk having your whole case thrown out for good if you sued under the wrong writ. Also, there was a certain formality to each writ and certain language you had to include in the corresponding complaint. All this is largely irrelevant now, however.
That's the difference as best I can remember it. I could, of course, be incorrect so doublecheck that against what your prof says. hth