Time is a strange thing. It's like those stalagmites you find in large caves: It's constantly dripping, but you can only see what is accumulated on the bottom. The stalagmite; the past. If you believe in determinism then you believe that there is another peg from where it's dripping down. The stalactite; the future. But no one has ever seen the stalactite. Some claim to have caught a glimpse, of course, but the whole future has defied scientific imaging efforts. And that's really something, since predicting the future is basically the only reason science exists.
Knowing the future
So, what exactly can we find out about the future? We can only approximate larger events and consequences. The more details we need, though, the harder it gets to determine what will happen. However, if you simplify, you leave out details that might be important. This is apparent in weather calculations: The meteorological system is so complex that predicting the weather at a specific location more than 5 days ahead of time is mere guesswork. Wow. 5 whole days! Demographs can find out which party a group of people are gonna vote for. By asking them beforehand. And even with that knowledge it's quite unpredictable. We can find out when a meteor will hit the face of the earth. If we've seen from where it's coming and how fast it's going beforehand.
Not quite the best images of our stalactite, so far. Is knowing the future possible at all? First of all, the following question arises: If you know what you are going to do, are you still going to do it? What if you don't do it? Did you change the future? How could you produce a different future, by knowing "the future"? Are you perhaps going to do that thing just to prove you were right? At the heart of these questions however is actually the question of free will, which I've covered over here (short and sweet answer: our past determines our future). So, sometime in our past we must have acquired knowledge of the future in order to know the future. In a world without time-travel, that's just not possible.
But how do those weather predictions work then, you ask? We use models for our predictions. Models are a simple description of the status quo and the assumed rules for the development of the system. With these two fixed inputs, we calculate how the real system would develop if it were equivalent to the model. So, predicting the exact future of the earth would require a system as large and complex as the earth itself. And this would still not be faster than the real one, because of the speed limit in our universe. This has even been proven scientifically, ironically. It's called the Entscheidungsproblem (although, it was conceived in a completely different context).
So, even if it's predetermined, we can never know the future beforehand, because it would defeat logic.
Now, that we've covered knowing the future, let's get to the actual philosophical question of whether the world is deterministic.
Science' one premise is that there's always a reason, a cause. But what if there isn't? What would it mean if there wasn't a predetermined course for the future? Then, by casual interpretation, some things would be left to chance. But what does chance really mean scientifically? It means, we cannot find a rational, scientific explanation for the given event happening. In other words, living in a non-deterministic world means that we cannot fully explain why things happen the way they do. That we cannot find underlying rules or principles for the events we observe.
How can you check if the world is a deterministic one without finding out the future? This one, we've already found out a long time ago: Record the past and try to make sense of it, by finding underlying rules and principles. This is what lead to all the scientific discoveries that lead to the theories of how our world works.
However, so far the theories on the most basic level, the quantum theories, suggest that events on this level can only be associated with probabilities and it's never possible to predict one outcome with certainty. The question is whether these probabilities are a physical reality (which would be fatal for determinism) or hide the yet unnoticed bustle of even more fundamental structures.
The crucial thing is that absence of evidence isn't evidence of absence. So, even if we had tried to find explanations for quantum events for a hundred years without success, we couldn't ever be sure whether there are none or if they've just eluded us thus far and we would keep searching even if there are none or they are out of our reach.
This battle with nature is still ongoing and, as we cannot know the future, your bet is as good as mine. I for one believe that there are structures that are impossible for us to detect and make sense of at this scale, since even the act of measuring the exact location of a "particle" disturbs it so much that the information is basically useless. However, even if we uncovered all the rules, being able to explain everything doesn't lead to being able to know the exact future, because time is a bitch and is as unskipable as it is unstoppable. So, even though there is no real free will, we still won't know exactly what we're going to do when it rains tomorrow. If it does.