What is the difference between these two phrases?
"He was going for a morning walk every day"
"He would go for a morning walk every day"
He was going for a morning walk every day. Incorrect
This sentence seems to be about a past routine, mainly because of the adverbial every day. But the past continuous tense is not commonly used for talking about routines. Usually, we use the present simple for that but the past simple works for past routines that may or may not happen in the present (the past simple is for past completed action - it really doesn't say anything about the present). The problem is with the time phrase - every day, it suggests a routine or habit. This is an example of where tense supplies vital meaning (it's call usage) to a sentence, and in this sentence the past continuous conflicts with the adverbial phrase, every day.
He went for a morning walk everyday. Correct - this is how we usually talk about past routines.
Next sentence: He would go for a morning walk every day. Correct
This is either the past participle of the auxiliary will with base verb go, or it is a modal used to form the subjunctive mood. So as a bare sentence, it has two possible interpretations. If it is about the past as in, He lived in countryside, and would go for a walk every day, or prefaced with In the past, he would go... It's meaning is the same as the past simple He went... but it feels casual and a little more distant in POV, and it hangs there a bit, as if there is more coming, but it is not incorrect.
If would is a modal (subjunctive), the sentence is then about something imagined, not something real. This use of would also suggests an incomplete conditional form where would suggests a possibility of imagined action, as in, He would go for a morning walk every day, if he were able. Or perhaps, in another context, the subject is imagining a situation in which He would go for a morning walk every day. It could be an imagined past or present:
Past: If I was still living in the countryside, I would go for a walk every day.
Present: If I was living in the countryside, I would go for a walk everyday.
The subjunctive mood is pretty complex and usually requires more context - notice that these sentences use two clauses to clarify their meaning. But sometimes that context can be provided by surrounding text where the sentence, I would go for a walk every day, has complete meaning.
That's why context is important for a sentence like He would go for a morning walk every day., because as a bare sentence its meaning can be taken in both a past time frame (he went...) or in the subjunctive as an imagined past or present.
He was going for a morning walk every day.
He would go for a morning walk every day.
Both the sentences are grammatical, but there's a difference between their meanings.
The first sentence is in the past continuous. You can use this tense to refer to something that you did again and again in the past. The sentence means that he used to go for a morning walk again and again every day. The sentence is grammatical though it's odd or unusual. You usually go once, not several times, for a morning walk every day.
You can use would instead of "used to" to refer to something that you did regularly or habitually in the past. The sentence is quite clear. It was his habit or routine to go for a morning walk every day; of course, it conveys the sense of going for a walk once, not several times, every day.
He would go for a morning walk every day suggests only that daily walking was his habit. When or how or why or even whether he stopped going for a walk does not matter.
He was going for a morning walk every day means exactly the same, up to the point that he stopped going for morning walks.
He was going for a morning walk every day demands an explanation for why he stopped doing that… as for instance, He was going for a morning walk every day until he died.
Contrary to what other users have written, the meaning can indeed be identical in some contexts. Here are some senses in which they diverge.
Either "he was going" or "he would go" can mean this:
Over a period of time in the past, he went for a morning walk every day. It was his regular routine then, but it no longer is.
However, only "he would go" can mean this:
He does not go for a walk every day, but he might do so in some hypothetical scenario.
And only "he would go" can mean this (but it only occurs in literary English):
There was a period of time when he did not go for a walk every day, but later on he did go for a walk.