Hello, Briana.

If you break words down, it will help you out to understand the difference of similar words

Even though I am a native Korean, I normally use this method when I am confused with them.

So let's break it down.

1. 사은품 : 사은(an expression of gratitude) + 품(=an Item)

You can use '사은' as a verb which is '사은하다.'. It means 'express gratitude'

(*Not many people use this verb nowadays though.)

Basically '사은품' is an item that somebody expresses gratitude through giving it.

In the commercial field when you purchased something they thanked you. So they gave you another gift.

That's '사은품'.

2. 증정품 : 증정(=giving something away) + 품(=an Item)

You can also use '증정' as a verb which is '증정하다.'. It means 'give away.'.

I've seen a number of people using 'giveaway' on the internet to promote their website or goods.

I guess that's equivalent to '증정품'.

Basically it's similar to '사은품', but it's a bigger version of it, because whether you buy something or not it can be given.

edited :

I have the same opinion as Bulabi except for "다음 달".

 It should be "다음달" without any space.

---------------

This is based on the dictionary '고려대한국어대사전'.

'국립국어원' says differently. 

Joo Hyung made a good point of it, and yet I'd like to add a few things more here. 

First of all, you need to understand that space among words can make the difference of a meaning when it comes to Korean.

I think you might be confused with '잘못' and '잘_못'.

If you want to put the verb '하다.' after '잘못'. it's without any space like this. '잘못하다.'

This is  one word. This means 'S/O does s/t wrong'. 

I guess that's why the website said to you, '하다.' can be used only. (well '잘못되다.' is possible too, but it's just about active or passive.)

However, if you want to put another verb after '잘못' except for '하다' or '되다', you need to make space between the two words. For instance : '잘못_찾다.(S/O searches s/t wrong)', '잘못_타다.(S/O rides s/t wrong)', etc.

On the other hand, '저는 글씨를 잘 못 써요.'  ('I am bad at handwriting.') needs to make space among the three words like this. '잘_못_써요.'. Because of the space its meaning can be changed.

Let's make some examples with 쓰다(Write)

1. 잘못_쓰다. (S/O writes s/t wrong)

2. 잘_못_쓰다. (S/O is bad at handwriting)

Fyi , S/O or S/T I refer to is 'someone' or 'something'