Small changes in a global average temperature are extremely important, because we don't ever experience "average" conditions. The frequency distribution of temperature can be roughly approximated by a normal distribution and the full range of the distribution (i.e. "natural variability") is large, relative to projected changes in the mean.
However, even a small shift in the mean of the distrbution results is much more dramatic changes in the tails of the distribution (i.e. "extreme" conditions). These changes include experiencing moderately hot weather more frequently and having the extremely hot weather be much hotter than it has been in the past. Our organization has studied this in detail for a number years at local scale. We typically see events that were 1 in 30 year heat waves in the historic climate (1970-2000) which are projected to be on the order of 1 in 5 events for the future. The details vary depending on what part of the distribution you're interested in (90th/95th/99th precentile), but the trend is always the same: more hot weather and hotter hot weather.
See this figure from the IPCC's Special Report on Extremes (SREX) for a good illustration if distribution shifts.
The last 20 years or so of climate science have focused on means, mostly because we haven't have the computational resources to study climate at high resolution (both spatially and temporally). That has been changing fast in the last few years and we're likely to see a lot more analysis and research on extremes in the future.