"I am a corporately filed LLC".

No such thing. Either the business is incorporated as an S-Corp or C-Corp, or it is an LLC, the latter of which is a disregarded entity indicating you'll be reporting business income/expenses/assets on a SCH C as a part of your personal return.

If the business purchases the travel trailer that makes it a depreciable business asset. You still report personal use, and that personal use will affect your deductible expenses as well as depreciation, based on percentage of personal use.

If "you" purchase the travel trailer, it is NOT a business asset. Expenses for it's use will NOT be reported on SCH C. Instead, they get reported on the SCH A subject to the 10% AGI rule for itemizing your deductions. With this method, you'd claim your vehicle expenses in personal income under the Deductions & Credits tab in the Job Related Expenses section. Those expenses, along with all your other itemized deductions would need to be "at least" 10% of your total AGI in order to be deductible.

I suggest you utilize the first method and the business purchases the trailer.

Note that with either method, you MUST keep detailed records of your mileage when there is a mix between business use and personal use.

In some states, who purchases the vehicle (you or the business) doesn't matter. It's how you treat it that first year of ownership that determines if it's a business asset or not. For example, if you purchased the trailer years before you started your business, and wanted to make it a business asset, you can just "do it' that first year. But it's value as an asset will not be what you paid for it years ago when you purchased it. The business value for depreciation would be its FMV at the time you placed it in service for the business.

If it is a disregarded entity on Schedule C, the IRS doesn't really acknowledge that the LLC exists.  That means you can purchase it in your own name if you want.  However, that MIGHT void any liability protection that the LLC has.

So, with that being said, is there any difference in the "business" purchasing the TT versus me personally?  The first gentleman seems to think there is a difference, your comment eludes to the fact that there may not be?.?.?

"You" and "the business" are indistinguishable because the LLC doesn't exist in the eyes of the IRS, in the same fashion that "you" and "the business" are indistinguishable if you didn't have the LLC but did have business income reported on Schedule C.  There might be other issues, issues not income tax related, that favor buying an asset in the name of the LLC.

I was just stressing things from a tax view, not from the legal/liability view. But from that latter aspect, if your business is of a type that is prone to lawsuits, you purchasing the vehicle does not protect it from liability if you claim a business use for it. But it does make it more work for a plaintiff.
In my single member LLC I purchased my truck in my name, but I list it as a business asset since I use it 100% for business. I'm not concerned about someone suing me in my business. If they did, they can still get my truck or have a judge force me to sell it if there was judgment against me. But when I purchased my truck I did so with the intention of using it for business only. At the time (over 10 years ago) I wasn't aware I could purchase the truck in the business name. If I had been, that's what I would have done. If I had done that, then when I sell the vehicle it would have been a business sale. But as it stands now, it would be a personal sale. I'm fine with it, because I generally drive my vehicles until they lose parts on the road to the point they won't run anymore and have no value anyway.