Integrated tube amp vs Tube amp?

Integrated amps are usually purchased in order to save space or compromise in some fashion. If you like the very distinctive sound of McIntosh, then definitely get a tube amp. Tube amps have a real musicality and realism hard to achieve with solid state.

I have a McIntosh C2700 with a MC275. The music the pours from my speakers is so easy and effortless. It's like liquid gold. I never ever get listening fatigue with this combination. I did listen to a C2700 with a MC462 at the dealer. Fantastic combination if you want a stronger or bass presence with more authority. 
The Tube pre-amp / SS amp combination absolutely has a different sound characteristic than all tube. I went with the tube amp because I just personally favored it over the two. I really enjoy that lush warm blanket of sound it creates.  

Only one word of caution. Be wary of the tube rolling rabbit hole if you go all tubes. It is real and it goes on forever..... 

Usually the amp is powering speakers, and the speaker impedance comes into play at some point.
If they are super sensitive, then don’t bother with the amp… IMO.

What is the sensitivity and impedance profile (or rated and minimum ohms)?
That will help as a way to provide some useful recommendations.

For instance, if it is those big Moab’s, then get a big class-D.

I am using McIntosh 302 amp now. I wanted to experience “tube” sound and was thinking of trying “horn” sound also by getting a pair of Klipsch Cornwall 4 s. Currently have Goldenear Triton 1s and Ohm 5 Mk2s.

In most cases, I would say yes - you'll get more performance from a pure tube amp. Tube integrated amps often have very simplistic preamp & volume stages in order to fit into the same box & budget. Simplistic is fine, but...if you hear an ARC Reference 6 or VAC Master, you will know what a big box pre can do (neither of these will come close to fitting into an integrated). The power amp section of an integrated could also be compromised a little for those reasons. Having to use an extra set of interconnects sucks, but still worth it imo. 

The biggest stopper to most people going to separates is having to buy 2 components up front instead of one - but you've already got the preamp. Go for the tube power! You'll also gain more flexibility down the road - sometimes the brand-matching pre & power aren't actually the best sonic match in your system, to your ears.