Spring and autumn are the most obvious choices but each season has its advantages and disadvantages.Spring is of course nice for the cherry blossoms but the downside is that it's still cold and most of the other nature is still dead and ugly, because the cherry blossoms are one of the first to bloom (they bloom after the plum blossoms which are actually more vibrant).
Summer is humid and June/July is rainy season, but there are many good about summer as well. There are so many festivals in the summertime and all the nature is pristine and beautiful so if you visit gardens or other outdoor sites, they'll be at their best. There are also flowers blooming which makes things even better.
Autumn is of course nice for the leaves. Unlike spring, autumn also has the advantage that the gardens and nature still look nice because they haven't died yet. Plus, the colored leaves will enhance the nature where there are deciduous trees, assuming that you don't come too early/late to see them. It's also not too hot and not too cold.
Winter is probably the worst for sightseeing, simply because the nature and gardens are all dead and it's cold however, if you go north or to places along the San-in Coast, getting to see temples, shrines, gardens, etc. with snow is a really special treat! It's also considered to be the best time to visit onsen. Even though you cannot climb Mount Fuji, the clearest days tend to be in winter, so it's good for photographing the mountain. Obviously if you have an interest in skiing or other winter sports, that will add to the intrigue of winter. The Sapporo Snow Festival is also held in winter. (to be fair, there are festivals in all seasons at some point, but summer has the most by far and they tend to be larger in scale)
If we just rank seasons, I suppose autumn is my favorite because it has very few downsides but in terms of REAL experiences, it depends on what you do. For example, I cannot say that seeing autumn foliage in Kyoto is necessarily better than going to a major summer matsuri (Nebuta Matsuri, Awa Odori, etc.). I also couldn't say that seeing cherry blossoms is 'more Japanese' than going to a matsuri. All of these are equally authentic experiences and very worthwhile.
These things need to be taken into account. There are many experiences that do not change much regardless of season, so you just have to decide what experiences are most interesting to you.
