What is the difference between "if" statements with "then" at the end?

Here's a quick tip that is not directly related to your question: in Ruby, there is no such thing as an if statement. In fact, in Ruby, there are no statements at all. Everything is an expression. An if expression returns the value of the last expression that was evaluated in the branch that was taken.

So, there is no need to write

if condition
  foo(something)
else
  foo(something_else)
end

This would better be written as

foo(
  if condition
    something
  else
    something_else
  end
)

Or as a one-liner

foo(if condition then something else something_else end)

In your example:

something.meth(if val == 'hi' then 'hello' else 'right' end)

Note: Ruby also has a ternary operator (condition ? then_branch : else_branch) but that is completely unnecessary and should be avoided. The only reason why the ternary operator is needed in languages like C is because in C if is a statement and thus cannot return a value. You need the ternary operator, because it is an expression and is the only way to return a value from a conditional. But in Ruby, if is already an expression, so there is really no need for a ternary operator.

then is a delimiter to help Ruby identify the condition and the true-part of the expression.

if condition then true-part else false-part end

then is optional unless you want to write an if expression in one line. For an if-else-end spanning multiple lines the newline acts as a delimiter to split the conditional from the true-part

# can't use newline as delimiter, need keywords
puts if (val == 1) then '1' else 'Not 1' end

# can use newline as delimiter
puts if (val == 1)
  '1'
else
  'Not 1'
end

The then is only required if you want to write the if expression on one line:

if val == "hi" then something.meth("hello")
else something.meth("right")
end

That brackets in your example are not significant, you can skip them in either case.

See the Pickaxe Book for details.