Both the market and the Shrine are well worth a visit. The Shrine includes extraordinary beautiful buildings, many different aged stone lanterns and numerous red and white plum trees that all breathe ancient times. Not strange when you learn the Shrine was built in 947. Both during blossom season as during fall the Shrine grounds should be a splendour of colour.
Some would say the market on the 25th is a flea market, but that's not entirely true. The market offers almost the same sort of antique items as Tō-ji, and these are real treasures mostly. There are also quite some food stalls, kimono, tools, toys and plant sellers and a few handcraft booths to visit. It is even possible to find exact the same sellers and their wares as at Tō-ji. This could be a downside if your looking for a completely different offer, but it could also be an upside. Since it's possible to visit the one if you missed the other and maybe you didn't buy that one item you actually should have… you're not offered a second chance like this very often! There is a saying that connects both antique markets even ''Fair weather at Tō-ji market means rainy weather at Tenjin market". A saying that portrays Kyoto's fickle weather.
Lucky for me I chose to visit the markets in April and the weather was great on both days.
I couldn't say which of the antique markets would be my favorite. They are both great. Kitano Tenman-gū buildings and grounds are maybe more charming but the downside is accessibility. If you're not a really early starter, you might look at the frustrating aspect you have to stand in such a long queue that you have to let pass three busses before you can finally board (a very crowded!) one. And you could also look at the same fate leaving the shrine grounds. Tō-ji, fortunately, is just a nice stroll away from Kyoto Station, so you don't really need to take the bus. So if times do get busy, you have a great alternative. I know this is my favorite tip, but I'm gonna say it again: start early! If this was ever a good tip for an activity in Kyoto, now it is.
Website to Kitano Tenman-gū, in English.