Il y a and depuis have different meanings. Depuis refers to a situation that has been true since a certain time in the past (past continuous), whereas il y a refers to a specific time in the past (past perfect). There is a similar distinction in English: depuis = since, il y a = ago.

Depuis une semaine, il n’a pas neigé. It hasn’t snowed for a week.
Depuis une semaine, il neige. It has been snowing for a week.
Il y a une semaine, il a neigé. It snowed a week ago.

When depuis is followed by a time interval, it can be combined with il y a (“depuis il y a une semaine”), but il y a is frequently omitted.

Depuis can also be followed by an event (“depuis son départ”, “depuis qu’il est parti”). Il y a can only be followed by an interval, some other preposition must be used to refer to an event (“lors de son départ”, “lorsqu’il est parti”).