The pump's major selling point with most people is convenience. Instead of six shots every day, one every three. If you're doing carb counting and corrections, it can do the math for you.
Some of its other advantages include being able to set a background insulin rate that more closely matches the amount of insulin you need at different times of day than Lantus or NPH is likely to do, being able to reduce or increase basal insulin quickly, and being able to get very very small doses reasonably accurately.
The pump's two biggest problems are cost (generally at least 2000 dollars per year more than shots, although insurance may pay all of it) and skin irritation. Some people also do not like being stuck to something all the time and feel more conspicuous. There is also a risk involved if the pump breaks and no insulin goes in; blood sugar can go high really fast and ketosis can develop. This is a pretty low risk if you check blood sugar often and change your site if you see highs.
The CGMs only strength is having a good idea as to what your blood sugar is doing all of the time.
That enables you to make smarter decisions about everything involved in your diabetes care, if you pay attention. It also allows you to treat high blood sugar sooner and low blood sugar sooner, which may prevent them from becoming as severe as they would otherwise.
If you use the medtronic system, you also can turn off insulin with lows- it can somewhat treat a low without you touching a single button.
The cons of the CGM are cost and skin irritation/pain. There is also a risk that you will mess up because of trusting it if it gives you an invalid number.