In Word, a table always has a paragraph mark just below the last row (you
can see it if you turn on nonprinting characters by clicking the ¶ button).
If you put the cursor just to the left of the paragraph mark below the first
table, and paste in the second table, they'll automatically become one
table.
Alternatively, if you paste the second table after the paragraph below the
first table, you should be able to delete the empty paragraph between the
tables to combine them that way.
If the tables have different column widths, combining the tables won't
automatically make them the same. It can sometimes be quite hard to get them
to match. One way is to convert the table to text, with columns separated by
tabs, and then convert back to a table.