From researchgate: https://www.researchgate.net/post/What_is_the_actual_difference_between_1st_order_and_higher_order_logic

What is the actual difference between 1st order and higher order logic?
Yes, I know. They say, the 2nd order logic is more expressive, but it is really hard to me to see why. If we have a domain X, why can’t we define the domain X’ = X u 2^X and for elements of x in X’ define predicates:
SET(x)
ELEMENT(x)
BELONGS_TO(x, y) – undefined (or false) when ELEMENT(y)
etc.
Now, we can express sentences about subsets of X in the 1st-order logic!
Similarly we can define FUNCTION(x), etc. and… we can express all 2nd-order sentences in the 1st order logic!
I’m obviously overlooking something, but what actually? Where have I made a mistake?

My answer:

In many cases one can reduce a higher order formalization to a first-order, but it will come at the price of complexity of the formalization.

For instance, formalize the follow argument in both first order and second order logic:
All things with personal properties are persons. Being kind is a personal property. Peter is kind. Therefore, Peter is a person.

One can do this with either first or second order, but it is easier in second-order.

First-order formalization:
1. (∀x)(PersonalProperty(x)→((∀y)(HasProperty(y,x)→Person(y)))
2. PersonalProperty(kind)
3. HasProperty(peter,kind)
⊢ 4. Person(peter)

Second-order formalization
1. (∀Φ)(PersonalProperty(Φ)→(∀x)(Φx→Person(x)))
2. PersonalProperty(IsKind)
3. IsKind(peter)
⊢ 4. Person(peter)

where Φ is a second-order variable. Basically, whenever one uses first order to formalize arguments like this, one has to use a predicate like “HasProperty(x,y)” so that one can treat variables as properties indirectly. This is unnecessary in second-order logics.