Hi Kevin,
Similar used when two things are more or less the same without being identical. For example, he has a similar point of view to her.
Analogous is used to compare 'relativity' of two or more things. It can also be used in a more formal way. For example, the wings of a fly, a moth, and a bird are analogous because they have a common function which is to enable flying.
You would rarely hear native speakers use analogous in day to day conversation.
Hope this helps!