SQL comes from relational algebra.
One way to look at the difference is that JOINs are operations on sets that can produce more records or less records in the result than you had in the original tables. On the other side WHERE will always restrict the number of results.
The rest of the text is extra explanation.
For overview of join types see article again.
When I said that the where condition will always restrict the results, you have to take into account that when we are talking about queries on two (or more) tables you have to somehow pair records from these tables even if there is no JOIN keyword.
So in SQL if the tables are simply separated by a comma, you are actually using a CROSS JOIN (cartesian product) which returns every row from one table for each row in the other.
And since this is a maximum number of combinations of rows from two tables then the results of any WHERE on cross joined tables can be expressed as a JOIN operation.
But hold, there are exceptions to this maximum when you introduce LEFT, RIGHT and FULL OUTER joins.
LEFT JOIN will join records from the left table on a given criteria with records from the right table, BUT if the join criteria, looking at a row from the left table is not satisfied for any records in the right table the LEFT JOIN will still return a record from the left table and in the columns that would come from the right table it will return NULLs (RIGHT JOIN works similarly but from the other side, FULL OUTER works like both at the same time).
Since the default cross join does NOT return those records you can not express these join criteria with WHERE condition and you are forced to use JOIN syntax (oracle was an exception to this with an extension to SQL standard and to = operator, but this was not accepted by other vendors nor the standard).
Also, joins usually, but not always, coincide with existing referential integrity and suggest relationships between entities, but I would not put as much weight into that since the where conditions can do the same (except in the before mentioned case) and to a good RDBMS it will not make a difference where you specify your criteria.