Is there any difference between writing a covered call or shorting a naked put? I just cannot see why someone will do two trades (covered call) rather than a single trade (write put option) to get a similar profit/loss profile.

Of course I understand that if there is a premium differential, one might favor one over the other.

Well beyond the premium difference (which can be considerable) the question is whether you want to own the stock in a flat market or not.

If yes, then a covered call is preferrable, as it gives you premium and you still own the stock. If not, then a naked put is preferrable as it gives you premium and you do not own the stock!

There is also an access difference. Some brokers will not permit naked option writing for traders without significant experience. Since a covered call does not involve invasion of margin that is typically allowed by accounts and traders who have little experience in the markets.

Naked option writing comes with significant margin requirements and can lead to "surprises" of the unpleasant kind.

I disagree with the premise that cover call and naked put have similar P/L profiles, due their opposite assignment risk.

A covered call is a way to purchase stock at a discount. (Buy stock, write/short call, with variations on Time Value and ITM/OTM.) There are two main, but opposite risk here:
1. That you want profit from a slight drop in the PPS, by getting the call exercised, but but if the stock falls below the strike you are holding without a hedge and start loosing money below BEP.
2. The other is that you want to buy the stock at a discount, and sell OTM calls each month for income, but the stock moves above the strike and you get exercised. This will reduce the profitability of this strategy due to additional transaction cost.
Both strategies have merit and manageable risk profiles, although I would be uncomfortable in the current environment with the prospect of holding an unhedged stock position.

A naked ITM put anticipates that PPS will rise above the Strike, and a naked OTM put anticipates that the PPS will not fall below BEP. The exercise risk it that you must purchase stock at the strike price, at a time when the PPS is already lower, resulting in a loss.

A short call/long put (in combination simulates short stock position) presumes falling prices vs. a long call/short put (together simulate long stock position) presumes higher prices.

Yep - covered calls are a nice strategy in a mostly-sideways market, especially if you already own the long underlying and are comfortable with giving up some part of the appreciation if the stock runs.

HOWEVER, in today's market holding that long could be nasty; ergo, in today's market I'm not doing ANY covered call writing!

I'm also not shorting PUTs, as I sure as hell don't want to own a collapsed underlying at a nasty-high basis!