Which is better for everyday office work?

The HP EliteDesk 800 G3. Its quad-core i5, SSD and 8 GB of DDR4 make Windows 10 Pro feel quicker than the older OptiPlex 3040 platform.

Which has better connectivity?

The HP. It adds USB-C, DisplayPort and Wi-Fi. The Dell OptiPlex 3040 relies on standard USB and wired Gigabit Ethernet.

Do I need the optical drive?

Only if you still use discs. The Dell has one built in; the HP has no optical drive and only a single 2.5-inch bay.

Which is easier to upgrade?

The Dell's mini tower has two 5.25-inch bays and a standard internal layout. The HP is smaller and uses an external power adapter, so internal room is tighter.

Is the HP worth stepping up to over the Dell?

Yes for most shoppers. The faster CPU, SSD, USB-C and Wi-Fi are real day-to-day gains. Pick the Dell only if the optical drive or full-size bays matter.

Are these good for gaming?

No. Both use integrated Intel graphics. They handle office work, browsing and video, not modern games.

HP EliteDesk 800 G3 Tower vs Dell OptiPlex 3040 Mini Tower

Buy the HP EliteDesk 800 G3 if you want a faster, more compact business tower with USB-C, DisplayPort and Wi-Fi built in. The Dell OptiPlex 3040 is the older, plainer machine: it has an optical drive and two 5.25-inch bays, but weaker connectivity and a slower platform. Both are refurbished office PCs, so condition and the seller's warranty matter more than the badge.

Choose the HP EliteDesk 800 G3 if you need USB-C, DisplayPort, Wi-Fi and a smaller chassis for a modern desk setup.
Choose the Dell OptiPlex 3040 if you want an optical drive and full-size drive bays for a basic office or media machine.

Prices

HP EliteDesk 800 G3 Tower: $40.00 at eBay (last seen Nov 10)
Dell OptiPlex 3040 Mini Tower: $199.99 at Newegg (last seen Nov 10)
Difference: the HP EliteDesk 800 G3 Tower is $159.99 cheaper than the Dell OptiPlex 3040 Mini Tower at the latest prices (Nov 10)