Are these two the same TV?

Yes. Both are LG's 65-inch B6 OLED with a 120Hz panel and 4K resolution. The listings differ in how much detail they show, not in the set itself.

Which is better for gaming?

They are equal. Both give you a 120Hz OLED panel, HDMI 2.1, VRR, G-Sync and FreeSync Premium, and LG's Game Optimizer.

The panel is the same, so longevity is the same. OLED care matters more than the listing: avoid leaving static logos or tickers on for hours and use the screen-shift and pixel-refresh features.

Is the B6 bright enough for a sunny room?

It is fine for most rooms but dimmer than high-end LCDs in direct light. Curtains or a darker wall behind the TV help during the day.

Do I need a soundbar?

The 20W built-in speakers are serviceable for news and casual viewing. For films and games, a soundbar or AV receiver gives a clear step up.

Should I step up to a C-series OLED?

Only if you want a brighter panel and a stronger processor. The B6 covers Dolby Vision, 120Hz and HDMI 2.1, so the gap is picture punch, not features.

LG OLED B6 65" 4K Smart TV vs LG B6 OLED 4K Smart TV

These are the same TV: LG's 65-inch B6 OLED, sold under two listings. Both give you a 120Hz OLED panel, four HDMI 2.1 ports, Dolby Vision and Dolby Atmos, and the same Alpha 8 processor. Buy whichever listing is the 65-inch B6 you actually want; the difference is the listing, not the set. Owners rate the B6 well for picture and gaming, with the usual OLED cautions about brightness in bright rooms and static-image care.

Choose the LG OLED AI B6 65-inch if you want the 65-inch B6 with the full spec sheet spelled out, including Wi-Fi 5, Bluetooth 5.3 and the Alpha 8 Gen3 processor.
Choose the LG B6 OLED AI if you want the same 65-inch B6 panel and features and are fine with a shorter spec list.

Prices

LG OLED B6 65" 4K Smart TV: $1,199.99 at Best Buy (checked today)