The future of weather is change. Easy enough to make that statement, but since the question brings people here periodically, let's think about it some more. I earlier mentioned the topic in Weather will still happen. Entirely true, but maybe not as helpful as it could be.
Let's go back and think about what we mean in talking about weather. Partly, it means 'not climate'. Itself also not the most helpful comment. But let's continue with both weather and climate in mind. My touchstone is "Climate is what you expect, weather is what you get." Whatever it is exactly that is happening around you right now, that's weather.
We can think a little differently and decide that our expectation -- climate -- is also part of what's happening. In that case, weather is the difference between our expectations and exactly what is going on. Since I live near Washington, DC, and it's the middle of July, I expect it to be hot. More precisely, from the Weather Underground's reports for Washington National Airport, I expect today's high to be 88 F (31 C). That's the climate for that station. If the actual high were to be 88, then as far as high temperature went, we were exactly on our climatology. Conversely, we could say that there was no 'weather' -- no difference between what we expected and what we got. It looks like the forecast for today is for the high to be 5 degrees below the climatology. So it seems more likely that we'll have 'weather' of 5 degrees cooler than normal for the high.
If we look day by day for here and other middle or high latitude locations, we'll find days that are 20-30 F warmer than usual (10-15 C), and days that are 10-15 C colder than usual. That gives us a sense of how large 'weather' is -- give or take 15 C from climatology. The figure depends on locations and seasons. In the tropics the weather variations, in terms of temperatures that is, are smaller than in the middle latitudes (if I remember correctly, 5 C, 10 F, is considered a big deviation from climatology in the tropics).
The difference between scale of weather (how many degrees away from climatology you get) in tropics and the middle or high latitudes helps us see what is happening to cause weather. In the tropics, the solar input is relatively constant day by day through the year. With similar solar inputs, you reach similar temperatures day by day, and year by year. The larger differences from climatology occur when you have some big system (large cloud bands, clusters of thunderstorms, and up to hurricanes) active. The thing which drives those big systems are temperature differences. The systems then try to flatten out the temperature differences. In low latitudes, they're more effective at this, so you see smaller variations due to weather.
Come to higher latitudes, where most people live, and you see some of those tropical systems coming up your way (hurricanes, typhoons, etc.) carrying that very warm, very moist air -- replacing the more moderate air 'native' to your location. Or, here in the mid-latitudes, wait a bit and get a wave of cold air coming down from the colder higher latitudes. On top of both, you have the fact that the amount of sun you get varies by a lot through the course of a year. If the only thing happening were the change in solar input, we could calculate the temperatures, and temperature changes, we'd expect (a climate estimate) using the simplest climate model.
Now let climate change enter the picture. Will it change the fact that solar input varies little in the tropics and tremendously at the poles? No. Will it change the fact that far more solar input is in the tropics than in middle latitudes? No. Will it change the fact that weather systems respond to temperature differences across the planet by trying to smooth out those differences? No.
Since the answers to all those (and a host of others that are related) is no, weather will still happen. A little more detailed:
we'll still see days/weeks/months, even years, where the temperatures run below normal.
we'll also still see temperatures run above normal.
In the mid-latitudes, those differences on a daily basis will still be 10-15 C (20-30 F)
Now an application of climate change to our daily observations. Let's say (to keep the numbers easy) that the climate change of the last century were a 2 F (1 C) warming at my location. Before that warming occurred, the expected high would have been 86 F. Our actual high of 83 F represents 'weather' of 3 degrees F below the former normal. Given the current, warmer, climate, it means today's weather is 5 F below normal. Anything odd about 5 F off normal for a day? Hardly. The record low is 15 F below the average low, the record high is 12 F above the average high. (Tamer numbers here than I quoted above because a) it's summer and the ranges are smaller and b) I'm more knowledgeable about the weather for Chicago, which is more variable than DC.)
Weather will still happen, and have similar magnitudes to the past. What changes, as climate changes, is the average and some more subtle figures. They'll be the subject of their own note later.