れる・られる + ままにDo as (one is told or ordered…), In accordance with, In compliance with, Just like

Both patterns use まま, but れる・られる + ままに expresses acting in line with orders or circumstances, e.g. 言われるがままにする ('Do exactly as told'). In contrast, standalone まま(に) is used in a wider range of contexts to express an unchanged state, e.g. 窓が開けたままになっている ('The window is still open'). The compound implies passivity; the base form, a persistent state.

Both convey doing something in accordance with something else, but differ in nuance. れる・られる + ままに implies doing something exactly as told or forced, with little choice, as in 指示されるがままにやった (I did it exactly as instructed). に沿って simply means in line with a tangible or figurative standard, as in 点線に沿って (along the dotted line) or スケジュールに沿って (in line with the schedule), without implying lack of control.

Both convey the idea of doing something in precisely a certain way, but differ in nuance. れる・られる + ままに implies passivity or being compelled by external forces. For example, 言われるがままにやった ('I did as told'). とおり can be used similarly, as in 言われたとおりにやった ('I did it just as I was told'), but also appears in neutral contexts. For example, レシピのとおりに作った ('I made it exactly according to the recipe').

The passive form (Verb［れる・られる］) is used to show that an action was done to someone. For example, 言われた means 'was told'. When combined with ままに 'in that state', it can express following instructions exactly, e.g. 言われるままに買った ('I bought it just as I was told.').

れる・られる + ままに implies doing something in accordance with external directions, often with little control, as in 指示されるがままに (exactly as instructed). にしたがって also means in accordance with, as in 規則に従って (in accordance with the regulations), but does not necessarily imply lack of volition.

Both structures express doing something in a certain way. れる・られる + ままに is used when someone acts as they're told, e.g. 言われるがままに 'just as told'. ように can express acting of one's own accord, e.g. 間に合うように早めに出た ('I left early so I could make it on time.').