Is there a difference between a cleric and a celestial domain warlock

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Greetings FlameOutsideOfStaff,

Are you asking from a Philosophical standpoint?

As far as how the Characters are built and played, there are lots of differences:

• Spell Slots
• Ability Score for Spellcasting
• Spells Available to Learn
• How they interact with their God / Patron
• Weapons available to be used
• Etc.

What specifically are you looking to learn?

how they interact with god/patron as well as how commoners will view them

A cleric is typically very devoted and loyal to their god. A Warlock has made a deal, so loyalty and devotion isn't necessarily required.

Commoners, being commoners, likely wouldn't be able to tell the difference.

Here's a suitable analogy:

Cleric: "Praise be to Talos!"

Celestial Warlock: "A New Hand Touches the Beacon"

The cleric gets more spell slots and get the option to wear better armor. The warlock gets to pick a unique handful of abilities from the Invocations list. That's about it. 

The relationship with their source of power does differ. For Clerics it's more like a patreon; the cleric does good deeds in a certain domain—Light, Light, Arcane, etc—and the deity of that domain that they worship rewards them with some power. The Cleric doesn't necessarily ask for or expect the power, it comes from them doing the right thing.

For Warlocks, it's more of a contractual (or scam) type interaction—a celestial that represents a certain divine power will approach the Warlock with a proposition (or the other way around) and terms will be negotiated. Do this for that much power, don't to this, don't hurt these people, do hurt those people etc. Sometimes the Warlock will find a "backdoor" to access a divine beings power, like an ancient ritual or circle of power. This is more like the warlock finding a cheque made out to "cash" and being all "Well if you didn't want anyone claiming this power, you shouldn't have left this ritual of the pact divine laying around. I didn't do anything technically wrong"

As for how the average commoner sees them, that'd depend entirely on cleric/warlock in question. A kind and benevolent warlock who has made a coercive pact with Shar might seem more priestly than a brutal war domain cleric of Lolth.