One of the biggest misnomers of the game not only today, but of all time, is that losing the roll-off to choose who goes first decides the game. This is absolute fudgemuffins. It is true some builds take great delight in getting the first turn and consistently winning who gets to choose who goes first is advantageous. As such, there are quite a few abilities across many armies which influence this but its important to remember this is influencing who gets to choose. And consistently getting to choose is a very powerful option to have. This is very different from "first turn = win." Let's break this down.
There are very few moments when you can dictate what your opponent does. Things such as tank shock, rage USR and psychic powers are some of the few and this is because such control over your opponent is a very powerful tool. Choosing who goes first falls into this category as well. There are some very obvious cases when you want to go second (i.e. against reserve based armies so you do not lose two whole shooting phases) and if you have an army where this is a common tactic against you (i.e. Daemons, Jumper BA, etc.), you being able to choose for your opponent to go first is a large advantage to have. In the end, being able to choose allows you to adapt to whatever army, mission or board you are playing on and this is what is important. Whether or not your army prefers to go first, there should be instances when you want to go 2nd (if not, change your list!) and therefore choosing who goes first is more important than who actually goes first.
With that out of the way let's look at the advantages of going first compared to going second.
Going First
The most obvious reason going first is advantageous is you get to do everything first. Your shooting phase (all of them, not just the first one) comes before your opponent's and your assault phase comes before there's too. You are more likely to have more options in any given movement phase, particularly the 1st couple, as the board is less cluttered.
If you are a shooting base army you'll get to get in your shooting before your opponent which can often lead to less firepower directed your way. You're highly unlikely to wipe your opponent off the board (or get even remotely close) as 5th edition has made things pretty durable (vehicle damage chart, 4+ cover prevelance, etc.) but you are able to get that first strike in, execute alpha strikes more readily and suppress/disrupt your opponent's plans before they get to do anything.
As a list more geared towards getting across the board or to midfield, you have clear lanes in which to do so and are therefore unlikely to be blocked in your own deployment zone by fast opponents. This not only allows you to close the distance to your opponent but ensure your army has cover the following turn (i.e. smoke launchers) and is the most advantageous position possible based upon the terrain. Any ranged firepower you have can also suppress/disrupt your opponent.
What this all boils down to is as the player going first, you can set the tone of the game early. This isn't to say you control the game but you can strongly influence where the battle will unfold and have the chance to get that first blow in.
Going Second
There are of course huge advantages to going to second. The biggest being you get to see what your opponent does and react first. Now reacting to your opponent is bad, you need to be able to predict to an extent what your opponent is going to do (or wants to do) and be able to stop it. However, being able to see what your opponent does in a given turn before you means you have extra knowledge with which to work. The obvious downside of this is they get to move/shoot/assault before you.
This advantage is seen most during deployment and allows for such basics tactics as a refused flank (everything of yours is on on side) and split fire bases (split into two portions of your army, each equally far from your opponent), etc. You can ensure all of your vehicles and infantry will have cover during your opponent's first shooting phase and that your army is setup accordingly against your opponent (i.e. defense against alpha-strikes, far away from aggressive armies, etc.). This advantage is HUGE and often over-looked. I regularly give my opponent the first turn so I can see how they deploy. This allows me to better prepare my battle plan and requires no guessing as to what my opponent may do with their deployment.
The other major advantage of going second is you do everything last. Say what? Ya, being able to move after your opponent is important for securing objectives (and table quarters) and can ensure you win games in your movement phase. Your reserves also come in later which is often very important when they come in early. Getting reserves in at the top of Turn 2 is often leaving them out to dry and unable to take maximum advantage of what they can do but that half a turn later makes all the difference.
Seizing the Initiative
All of this is further balanced by seizing the initiative. That little 1 in 6 chance of taking advantage of an aggressive, going first deployment can really affect how a person deploys or changes the tides of battle. It's fairly uncommon, and even in lists with seizing advantages, isn't going to happen more often than not. This is an excellent balancing factor which can make giving your opponent the first turn a great proposition though if you are banking on seizing, it can be a dicey proposition if you fail.
Conclusion
Going first is great. Going second is also great. It all depends upon the matchup and the mission which results in being able to choose who goes first being very important. I'd much rather win the roll-off to be able to choose than go first all the time. In the end you need to weigh up the options of being able to react to your opponent and moving onto/tank shocking objectives last (going second) compared to getting that first punch in (going first). This will vary a lot depending upon your opponent and the mission so have some basic outlines of when you want to go first and when you would prefer to go second. That way when you win the roll-off, you can decide quickly without having to factor in a lot of information within a short time span.
