meaning - "Would" means something different in the past than in the future?

This usage of would is considered fairly casual, but the difference between your two examples is only in point of view: whether you're telling the story as your past self (I would often notice) or as your present self (I often noticed). Another common application of would in a similar context is for habitual actions:

I would go there every day after school when I was little.

Which is identical to:

I used to go there every day after school when I was little.

Generally the version with would only makes sense if a time is specified, or at least understood from context. So this is valid:

I remember when I was little. I would go to my grandma's every day after school.

But the second sentence by itself makes the listener wonder when the speaker is talking about. So generally speaking, these are identical:

I used to…
In the past, I would…