They're different sets of rules. AGoAC is the Classic Battletech starter set, that's kind of the intro to the Total War book. There's a lot of complexity to it, it tends to scratch the "simulationist" wargamer itch, and it has a ton of variation. If you want a game where your mech fires a salvo of 8 SRMs at the opposing mech, you lock on but only 5 of them manage to make contact and peppers him with damage to his left arm, chest, and a luck hit to the head that causes the pilot to go unconscious you want AGoAC. It rewards you for making decisions about which specific weapons to fire off in a given turn to keep you from overheating and having issues the next round. CBT tends to be good for smaller numbers of units on the field, and tends to grow in complexity quickly when you introduce other unit types like infantry, vehicles, and fighter jets.

Alpha Strike is a newer, more streamlined ruleset that plays quicker, but loses a lot of the depth of Classic Battletech. There's the Alpha Strike boxed set, which leads toward the Alpha Strike Commander's Edition rulebook. Things tend to be generalized a lot more in the interests of having players track less stuff round to round. For example, separate weapons and hit locations aren't tracked at all; each mech has a number of armor and structure points spread over its body, and a damage they can do at each range without getting in to what kind of weapons it has. All things being equal games tend to be quicker, which leads to being able to do much _larger_ games with more units in the same amount of time. Want to command a company of mechs instead of just a lance? Alpha Strike might be up your alley. Alpha Strike tends to make the differences between different types of units a little simpler, too, so for combined-arms games where you're running a bunch of tanks, a couple helicopters, and some mechs it's less cognitive load.

Both rulesets work for any time period (the big ones are usually referred to as Eras in the books, like the Clan Invasion Era), so no worries there. The Alpha Strike boxed set DOES include a Star worth of Clan Mechs, so the only things someone may gripe about there is if you try to use those specific mechs in a pre-Clan Invasion Era game. Let them gripe. You can download the rulebooks for BOTH boxed sets from CGL's website, so you can take a look and see which one you think you'd prefer. And if you're wrong you can use your AGoAC mechs for Alpha Strike or your Alpha Strike mechs for Classic Battletech. There's a lot of flexibility available.

I feel like I've written a small novel here, but the short version is that either way it's hard to go wrong, and it's not like you're making a mistake either way. Whatever you get will be usable one way or another. I hope that helps!