"Is this warning due to wiring or something, or because of heat build-up inside the globe?"

Generally it has to do with heat buildup and its possible effects on the socket and the ability of the insulation on the wiring to withstand this elevated temperature.

For recessed fixtures there is the additional concern about the fixture itself becoming hot enough to damage or ignite insulation in the ceiling, the framing or any nearby wiring.

For surface fixtures, as it sounds like you have, your options are: To risk a higher wattage lamp, bulb. This isn't a huge risk as long as the over-wattage is moderate, usually the result is lowered lamp life and eventual destruction of the socket, and/or wiring, in the fixture, the chance of a fire is tiny but significant. I have seen fixtures over-lamped to 200% of its rating and the result, after decades, was the destruction of the fixture, the box above, the wiring entering the box and the drywall above the ceiling. As bad as it was, a major repair job, there was no fire and the fixture was still running when we got there.

Another option would be to changing out the fixture itself for one rated for a higher wattage. Note that 60w is the standard for most fixtures. Look closely and pay a bit more when buying the replacement and you can find one rated for a 75w bulb and there are some rates as high as 100w.

You are correct in assuming that fixtures that tightly enclose the bulb/s with a glass or plastic cover tend to be more sensitive to heat buildup if over-lamped. Fixtures with shades that are open and allow air to circulate tend to handle over-lamping more gracefully. In an open fixture rated for 60w bulbs I have seen a lot of people put in 75w bulbs and have rarely seen any problems. Most of those in fixtures that were clearly over 40 years old.

In recessed fixtures a bulb that gives you more usable light can get you the brightness you want within the limits of the fixture and trim. A 'flood' bulb, PAR (Parabolic Aluminized Reflector) bulb projects the light down so a lower wattage bulb can get you more light.

In circuits that don't have a dimmer, fluorescents don't always get along with dimmers, compact fluorescent bulbs will give you more light while staying within the heat capabilities of the fixture. But some people don't like the light or performance characteristics.