Den nye app er mere moderne end den gamle, og den er også mere effektiv i hverdagen.

Because Danish often uses a kind of double definiteness when a noun is definite and has an adjective in front of it.

• appen = the app
• den nye app = the new app

So with a definite noun phrase like the new app, Danish uses:

• den
  • adjective + noun

The noun itself stays in its basic singular form: app, not appen.

In Danish, adjectives usually take -e when they are used before a definite noun.

So:

• en ny app = a new app
• den nye app = the new app

That -e ending is very common in definite phrases. The same pattern appears in den gamle.

Because app is a common gender noun in Danish.

Danish nouns have two genders:

• common gender → uses en, den
• neuter gender → uses et, det

Since it is:

• en app
• den nye app

you know app is common gender.

Danish can form comparatives in two ways:

1. with an ending, often -ere
2. with mere
   • adjective

For many longer adjectives, especially ones like moderne and effektiv, Danish commonly uses mere.

So:

• mere moderne = more modern
• mere effektiv = more efficient

This works very much like English more modern and more efficient.

End means than in comparisons.

So:

• mere moderne end den gamle = more modern than the old one

This is a very important comparison pattern in Danish:

• mere ... end ... = more ... than ...
• større end ... = bigger than ...

Because Danish can leave out the noun when it is already understood from context.

So den gamle here means:

• the old one
• more specifically, the old app

This is very natural in Danish, just as in English you can say the new one or the old one.

This is about normal Danish main clause word order.

In a regular statement, Danish usually has:

• first element
• finite verb
• subject
• other elements

But if the subject comes first, the order looks like:

• den er også mere effektiv

Here:

• den = subject
• er = finite verb
• også = adverb

So og den er også ... is the most neutral, natural order.

Because the word it refers to is app, which is common gender singular.

Predicative adjectives in Danish agree with the noun:

• common gender singular → effektiv
• neuter singular → effektivt
• plural → effektive

So:

• en app er effektiv
• et program er effektivt
• apps er effektive

In your sentence, den refers back to app, so effektiv is correct.

I hverdagen is a fixed, natural way to say in everyday life or in day-to-day use.

Here hverdagen is the definite form of hverdag:

• en hverdag = an everyday day / a weekday
• hverdagen = everyday life / daily life, depending on context

After i, Danish often uses the definite form in time and lifestyle expressions like this.

Også means also or too.

In a normal main clause, it often comes after the finite verb:

• den er også mere effektiv

That placement is very common and neutral. It adds another piece of information:

• it is more modern
• and it is also more efficient

Yes, absolutely.

• mere moderne end den gamle
• mere moderne end den gamle app

Both are correct. The version without app is shorter and sounds natural because the noun is already obvious from context.

Danish often avoids repeating a noun when it is clear what is being talked about.