Hm I also forgot about some other rounds and some more comments.
Flechette's are weaker against vehicles and spirits because of the terrible AP values. While against people, you don't really care if you're doing P or S after armor*, against vehicles and spirits it's all about defeating armor.
Likewise, APDS is generally more useful than Ex-Ex. It does the same damage in all cases you care about (+1DV -1AP is equal to -4AP on average against anyone with 4 or more ballistic armor). However it does a better job punching though armor. An APDS heavy pistol is 5P -5AP and that'll cut though a force 5 spirit's armor. The higher availability is annoying since you have to wait till play to get it.
Arsenal introduces new ammo options. Most are niche. Frangible is for not breaking stuff, Hi-C is for getting past MAD scanners, silver is for allergies to silver, subsonic is for being quiet. Capsule rounds deliver chemicals so that depends on what poison you pick. Tracker rounds are so you can be Batman and put a tracking device on a car via gun. AV rounds are slightly more expensive APDS rounds that do better against vehicles.
War has some new rounds. Anti-Tank is super APDS for big guns. RFID rounds are for letting yourself get IDed (which can useful for false flag operations). Sensor rounds shoot out sensors like you're Batman. Suppression rounds are to scare.
AP flechette and high-power rounds are the two DV increasing rounds in War. They are kindof hard to get, but if you're getting APDS, you might want to get them instead. AP flechette is +2 DV -0AP against impact armor. Thus it's +2.66 damage in most cases, double Ex-Exs or APDS's bonus. The -20% range can get annoying, but isn't a factor in most cases.
High-power rounds are +2DV -1AP. That's not as good as AP flechette's effective damage bonus, but is better against vehicles and spirits though APDS style rounds work best there still. In addition, you take a -2 penalty to shots with high-powered rounds. Recoil compensation most likely works against this -2 so it's a not much of a problem with SA and BF. It could get annoying with HV fire. If you can't compensate the recoil, the net damage gain is 1.66 which is meh. In addition, you have to mod your firearm to shoot high-power rounds. The mod uses up 2 slots which is annoying. Personally I'll just do AP flechette and if the range is too bad, get an extended barrel or improved range finder mods on the guns.