Novi madrac je udobniji nego stari.

It breaks down like this:

• novi madrac = new mattress
• je = is
• udobniji = more comfortable
• nego stari = than the old one / than the old mattress

So the structure is:

subject + verb + comparative adjective + comparison

Croatian is doing the same basic thing English does in The new mattress is more comfortable than the old one.

Because they are adjectives agreeing with a masculine singular noun in the nominative case.

The noun madrac is:

• masculine
• singular
• nominative because it is the subject

So the adjectives that go with it must match:

• novi madrac
• stari madrac

At the end of the sentence, stari stands for an omitted stari madrac, so it keeps the same masculine singular nominative form.

The base adjective is udoban, meaning comfortable.

Its comparative is udobniji, meaning more comfortable.

So:

• udoban = comfortable
• udobniji = more comfortable
• najudobniji = most comfortable

You do not usually translate udobniji word-for-word as comfortabler in English, but grammatically that is what it is: the comparative form of the adjective.

Both can be used in comparisons, but the structure changes.

With nego:

• Novi madrac je udobniji nego stari.

With od:

• Novi madrac je udobniji od starog.

Both mean the same thing here.

A simple way to think about it:

• nego often works like than
• od also introduces comparison, but it requires the following word to be in the genitive

So after od, you get starog, not stari.

The noun is omitted because it is obvious from context.

So stari here really means:

• stari madrac = the old mattress
• or more naturally in English, the old one

Croatian often leaves out a repeated noun when it is already clear.

So:

• Novi madrac je udobniji nego stari madrac
  is possible, but
• Novi madrac je udobniji nego stari
  sounds more natural because it avoids repetition.

Because here the adjective is standing in for a whole noun phrase: stari madrac.

• star is the basic predicate form meaning old
• stari here functions like the old one or the old mattress

Compare:

• Madrac je star. = The mattress is old.
• Novi madrac je udobniji nego stari. = The new mattress is more comfortable than the old one.

So at the end of your sentence, stari is not just saying old. It is replacing old mattress.

Because the sentence is not comparing age. It is comparing comfort.

• udobniji = more comfortable
• stari = the old one

If you said stariji, that would mean older.

So:

• nego stari = than the old one
• nego stariji would mean something like than the older one, which is a different idea

Yes, in a normal full sentence it is.

Je is the 3rd person singular present form of biti = to be.

So:

• Novi madrac je udobniji nego stari. = correct standard sentence

Without je, the sentence would sound incomplete in ordinary usage.

Yes, Croatian word order is more flexible than English, but the neutral order here is very natural:

• Novi madrac je udobniji nego stari.

You can move words around for emphasis, but that changes what sounds most prominent.

For example:

• Udobniji je novi madrac nego stari.

This still makes sense, but it puts more focus on more comfortable.

For a learner, the original order is the safest and most natural choice.

Because Croatian does not have articles like English a and the.

English says:

• the new mattress
• the old one

Croatian simply says:

• novi madrac
• stari

The meaning is understood from context, word endings, and sentence structure instead of articles.

So even though English needs the old one, Croatian can just use stari.

Because od is a preposition, and it requires the genitive case.

So:

• od starog = than/from the old one in comparative structure

But nego does not work that way. After nego, Croatian can keep the compared noun phrase in the nominative form:

• nego stari

So the contrast is:

• nego stari
• od starog

Both are correct, but they follow different grammar patterns.