I took our kids to the Armouries, it's an amazing museum with plenty to interest both ages, it's full of wonderful history and was so much more exciting than I expected. It's in the centre of Leeds so easy to get to. Hotels would prob be cheaper in Leeds because you have more budget brands available. Taking the half hour trip on the train to Harrogate takes you thru a bit of countryside, and yes Harrogate itself is a lot more scenic than Leeds.
There will be more to do in Leeds but Harrogate is much, much nicer. Public transport is decent enough in Harrogate and good in Leeds. How about spending Saturday in Leeds then catching the train to Harrogate for a hotel in Harrogate, then spending Sunday in Harrogate? Or vice versa?