Androids have the best starting physical stats and are completely immune to psionics. Their physical stats, particularly accuracy, will be surpassed by human soldiers that manage to live long enough, but their immunity to psionics and Brainsuckers can give them a place in your roster even after you have a roster full of experienced humans. Their high strength, immunity to brainsucking and psionics and the very unlikely chance of them going berserk due to their high bravery combine to make them perfect candidates to carry heavy weapons, since it's almost impossible to subvert an android to turn its weapon against you.
They are highly dependable and very durable (and some with a more macabre outlook on life would say, expendable. A human rookie is easier to replace than an android, but an android is easier to replace than an experienced human). Androids can go so far as becoming body shields to protect their other training comrades, because they are incapable of using training facilities so time spent in the medical bay doesn't disadvantage them. As such, androids make excellent shock troopers and are ideal for sending into UFOs first.
As androids are mechanical in nature, they are mostly ignored in the field of psionics, which means that though they cannot use it, they are also immune to it. That maxed out Psi Defense stat is just for show, as psi-capable enemies ignore them completely. Androids are also not considered legitimate targets for brainsuckers (though they still cause brainsucker pods to hatch). Brainsuckers seem to behave variably around androids, sometimes fleeing and sometimes charging up to their feet and then sitting there. If there is a suckable target near the android it won't hesitate to jump on them though. It is technically possible to get an android brainsucked if they leap in the way of a jumping brainsucker, but this happens incredibly rarely in practice.
Creating all-android squads or even using them exclusively for your ground teams has advantages and disadvantages. An all-android squad is less versatile, but has no weak links to exploit in its immunities. A brainsucker who makes it to a mixed squad can still brainsuck someone, while it's completely neutralised against an all-android squad. Similarly a Psimorph can't control the android members of a mixed squad, but if it can control any agents then it's infinitely more dangerous than if it can control none. A mixed force of androids with other races can use the android as an advance scout, putting to use its high speed and stamina to get around, toughness to weather surprise attacks, and spotting brainsuckers and psionics for the rest of the squad while remaining safe from them. In some terrain androids can screen the rest of the squad from brainsuckers by blocking a door or corridor, which is very important in TB mode, because brainsuckers are a much greater threat than in RT.
Maintaining good relations with S.E.L.F. is key to ensuring a steady supply of potential android recruits.