Why are non-vascular plants like mosses not considered Algae whereas seaweeds are considered algae?

I think the term "non-vascular plant" now includes algae, although I don't think that really goes to the heart of your question. The reason why mosses are considered "plants" and sea weeds are considered algae probably comes from the fact that there are certain characteristics shared by all land plants that seaweeds don't have. This primarily has to do with the reproductive cycle.

Both have a haploid and diploid stage but in algae, and in seaweeds, the two stages are separate plants, whereas land plants have the two stages essentially dependent on each other, usually with one stage as the dominant stage. Secondly, all land plants have reproductive organs that are protected by non-reproductive tissues. I don't think that is the case with most algae, including sea weeds.

Please double check this information with a reputable source. To be honest, I feel like I really had to find very subtle differences to answer your question, so who knows? Maybe someone has done some serious research and found that seaweeds are more like other land plants than mosses.  Anyway, hope this was helpful.

This is a very good question and not an easy one to answer.

Short answer..... Yes, it is based on DNA classification.  

Long answer......

Realistically, some members of any clade or other classification can easily be defined as either "plants" or "non-plants" depending on how loosely you want to define the terms.

Some algaes (brown, for instance) are very similar to the simplest of land plants (mosses, liverworts, hornworts).... as you mentioned.  They even have what is called a "stipe", which isn't technically vascular tissue, but it does transport some products of photosynthesis.

Classification is a tricky business and it is generally done by consensus.  Not everything will fit neatly into a classification system and there will always be little issues like this.  That is why they are constantly reclassifying things to get ever more specific.... hence the introductions of things like Domains and Clades fairly recently.

Think of it this way.... Life doesn't need to be classified, but WE would like it to be!