There are differences between males and females but this is mainly die to evolutionary traits. I'm not sure about reaction times but I can understand why one would believe males possessed faster reactions.
Basically every trait or difference between the sexes is the result of us evolving that way. For example, the colour spectrum;
Females possess the ability to see colours males cannot. This is not a joke, their eyes are actually better developed than males for the purpose of differentiating colours. This is because of the historical role of women in society.
In pre-civilization it was generally the female who acts as gatherers of herbs and plant-life while the male would be out hunting. Hence, females needed to develop good eyes for detecting minor variations in colours so they wouldn't end up poisoning them and the whole tribe.
Later on, as their typical gender role was in the home they were traditionally in charge of decorations and stylisation of the home or other buildings. This is why all the famous paintings by male artists had a rather small variation in colour. Take the Mona Lisa, very good but almost completely grey, brown and green right? This has contributed to females developing above average eyesight for colour differentiation.
For males, take the long held stereotype of direction and map-reading. As previously mentioned, it was the male who was out hunting, hence they developed a need for good direction and an accurate internal compass. Then, throughout history, males traditionally performed the 'outdoorsmen' roles and so continued to develop this spatial awareness.
In both cases, the other gender never received the need to develop that particular trait and so the quality was underutilised and consequently devolved over time. E.g: men didn't need a good perception of the colour spectrum to see the lion, and women didn't need to know where the cave was, seeing as they did not leave it as often or as far.
I will finish by saying that unfortunately, there are genes that determine abilities in genders, but there are none known that effect their ability to perform in modern society. E.g: the belief that men are better at maths is nonsense. This is just scientific fact.
