Everyone is probably aware that all pro players use a monitor. There is good reason behind this that I will explain to the best of my own understanding.
There are two types of measurements that matter directly in terms of reaction time. Response time and display/input lag.
Response time is the amount of time it takes a pixel on the screen to change from one color to another. This is usually represented by GtG (Grey to Grey). A high value here results in ghosting when the objects on the screen are moving quickly. The pixels do not change fast enough and you see a trail of the still changing pixels behind the moving object. Many monitors have around 5ms down to 1ms response time, 1ms being the best. TVs can have much higher response times due to the extra processing they do. You can expect to potentially see 15ms or greater response times for older TVs.
The other measurement, display lag, is the time it takes the source material signal to appear on the screen. This is the time it takes after a button press for the action to appear on screen or for the position of the enemy to update. This is a huge factor in being able to respond quickly to what is happening in the game. A good gaming monitor will be around 10ms or maybe even a little less. The best TVs can only get down to around 17ms with the vast majority of TVs being closer to 50ms or more. This is basically doubling the lag that many people already get from their internet. This is due to the extra processing that TVs do to improve image quality. Unfortunately this is obviously bad for serious gaming.
Overall display/input lag times for many TVs and monitors have been tested and listed here. http://www.displaylag.com/display-database/
A factor that some may be interested in if considering switching is the panel type within the monitor. TN panels are the most common and have the best response times generally. Another type that is becoming more common are IPS panels. These panels provide brighter screens and better viewing angles. TN panels can normally viewed within 160 degrees vertically and horizontally. IPS panels can be viewed at 178 degrees. This viewing angle is how far you can move to the side or up/down before the colors of the pixels start to shift because you are looking at them outside their optimal angle. This is personal preference, but I prefer IPS panels because of how close I sit.
For anyone interested, I got an Acer R240HY for $130 on Amazon. This is an IPS panel so the response time isn't perfect at 4ms, However, the overall display/input lag is around 9.6ms which is fantastic. The viewing angles are fantastic and the screen is very nice. Https://smile.amazon.com/Acer-R240HY-bi ... cer+R240HY
If you want to plug your xbox into a monitor and a tv at the same time for easy switching in between I have this hdmi splitter and it works well.
Https://smile.amazon.com/HD-102-Powered ... i+splitter
Feel free to post any questions and I will do my best to answer them based on what I know.