What are the pros and cons of suspension seats and suspension handlebars? The title says it all, but I’ll add more specifics to give a better understanding of my particular situation.

I am a on-road commuter with little-to-no dirt roads in my routes, many of the roads are VERY degraded and as such are very bumpy. My bike has 26x1.95 tires on it aired up to around 50 psi. I have no suspension other than my seat(spring) and seat stem, as well as on my handlebars. I happen to weigh approximately 225-250 lbs, if that matters. Additionally I do carry a messenger bag or backpack on most days, usually adding another 10-25 lbs. This question regards the pros and cons of the suspension seat stem and handlebars, although I’m open to extending it to other forms of suspension and other methods such as larger tires and tire softening. This is a general question but I would also like to know specifically what I should know about my bike, including upkeep and other things to keep in mind, particularly regarding the suspension I have on it.

tl;dr: I have a road bike with suspension seat stem and handlebars, I commute on degraded and bumpy paved surfaces, I weigh 225-250 lbs on average, I usually carry on-person cargo weighing approximately 10-25 lbs. What should I know about the suspension seat and stem and what other relevant information should I be informed of.
Suspension seatposts and stems have relatively limited travel--I think most suspension posts have about 20 mm of travel--so they're not so much intended to absorb big bumps as iron out road buzz. 

Most suspension posts and stems are heavier (perhaps not so heavy as to bother you), more expensive, and sometimes more finicky to set up.

You're already running pretty fat tires, although perhaps you could go up to 2.5" ones. According to the tire pressure optimizer it looks like you could use 40 psi in back and 24 psi in front, which would smooth things out a bit. There are some light-casing tires that are noted for giving a smoother ride (they deflect more readily around road irregularities), although I would hesitate to recommend them for riding over bad roads.

It's hard to answer specific questions about your bike without knowing more about your specific bike.