Philosophy and science are different in one respect: scientific theories can be tested by experiment, philosophy can’t.
What does this mean?
All the stuff other than testing is done the same. So, how much of this other stuff is there? How similar are they really?
Testing is the act of taking two (or more) theories with different predictions, and finding out which prediction happens by doing an experiment. The theory that fails the test (predicts something different from what actually happens) is eliminated, or changed to account for the unaccounted-for result.
Tests only happen when we have two viable theories. If one of them makes less sense than the other — has more holes/problems, or doesn’t explain as much as the other theory, or whatever — then the other is preferred by default, and no test to see which is better is necessary. (‘Cause if you can already see which is better…)
I say two theories, but it could be that there are more than two rivals on at one time. The thing is, that’s rare. It’s rare enough at the leading edge of science that we have one viable theory, let alone two, let alone more than two. Usually, when there are rivals, we can eliminate all but one or two by just using criticism. Tests only come in after we’ve done that.
So, what is there other than testing?
First, there’s coming up with a theory in the first place. We do this by guessing what might be the case, and guessing explanations for it. In other words: conjecture. (We do not induce theories from observation — though we can criticise our theories using observation.)
Then, there’s criticism. We criticise the theory to see if it makes sense, to see if it’s better than its rivals, to see if it explains what it purports to, and so on. We try to find problems with the theory.
When we find problems, we try to solve them. Either we will change the theory to account for the problems, or we’ll come up with a new rival theory that has fewer/less-severe problems, or we’ll discover the things we thought were problems are actually OK or explained already.
This process of criticising our theories and changing them to solve problems is intense. Or at least, it should be. Scientists tend to do it pretty well — they’re rigorous, or at least have a culture that encourages being rigorous. Philosophers… not so much. Half of them don’t even believe in objective truth, which can be a bit of a damper if you’re trying to find it.
Most people have the impression that philosophy is this wishy-washy, personal/subjective thing that you talk about to sound deep. Some people have an idea of what it is, but it’s mixed in with this wishy-washy conception of it. It’s no secret that most people barely know what it is; most philosophy classes start with the question, “What is ‘philosophy’?” (starting a history class asking “What is ‘history’?” would be absurd).
But philosophy is only different in this one way. To make good progress in philosophy, it needs just as much rigour as science (or even more, considering we don’t have testing to help us out).
Well… from the ‘testing’ difference, you could say another difference is that science and philosophy discuss different problems. Science is about the problems/theories regarding things we can test for (physics, chemistry, and so on), philosophy is about the problems/theories regarding things we can’t (morality, epistemology, etc.). But their methodology is the same (well, up until the point where you actually test stuff — which, by the way, doesn’t always happen for any given theory. A theory can be scientific without being tested; it just needs to be testable).
But it’s not just philosophy that has this similarity to science. All fields where one can make progress involves this conjecture-and-criticism process. And to be good at them, they all involve some degree of rigour.