If you get a good score on STEP ii or iii you should be able to score greater than 70% on the MAT, which is what the admissions tutors are on the lookout for. But I'd say the converse is likely to be true too. I think people who get 70% in the MAT should be capable of getting onto the 1/2 boundary for step and some quite a bit higher.
Points for the MAT being harder:
-Much less space to think in the MAT than in the STEP. You really have to rely on your intuition rather than sussing out an answer after trying out cases etc.
-You have to answer all questions on the paper, unlike STEP where you can choose the questions that suit your strengths.
-Time constraint can really mess you over, if you get hung up on one question you can't solve.
-many more questions, so if you want to get through the material you have to be quick at using nifty algebraic/ geometrical tricks to get through the material in the time.
Points for STEP being harder:
-I've found lots of the questions in STEP are an algebra grind; if you are very methodical and precise you can do well. But the algebra is really on a different level from Oxford's tests. However, I would say you still have to use just as many (perhaps slightly more) nifty tricks as the MAT.
-Questions are more interesting and are from a broader range of topics
-People you are up against would have been preparing since the start of the year, usually for a longer period of time than people prepping for the MAT- especially since there are more step revision materials than MAT
-You have to know what topics in maths you are best at
-More space to think, which may mean you have to think deeper. however, I think it's more likely that you are given more time to focus on fewer question because of the insane algebraic content of the exams , not because the exam requires more insight. Personally I think the two exams are the same with regards to insight required.
TLDR; STEP has a wider range of topics and much harder algebra than the MAT. MAT has a much worse time constraint and requires about the same level of mathematical (perhaps slightly less) insight/ tricks up your sleeve kinda thing.
People on TSR feel STEP is harder because of the wider range of topics and the harder algebra. But this gets conquered through preparation, and practicing on the many STEP past papers that are available. There are many repeated questions in STEP/ elements that are repeated so over time STep becomes a lot easier. With the MAT, there are fewer question papers, which means that learning the tricks expected by examiners is harder. Also, the earlier questions might be easier/ fairly standard, but the last few questions get quite diverse and quite hard to do in the time limit. You also need a score of 70%+ to be in with a good chance.