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Some medications such as birth control pills, Aspirin and estrogen therapies are known to be a cause of broken blood vessels on the face. They increase blood pressure, causing breakage of facial blood vessels and hence the redness common on the cheeks and around the eyes. Aging is also a cause of broken blood vessels. The red spots common on the elderly are not the color of aging skin. They are the result of broken blood vessels. As the human body gets older, it becomes fragile and prone to damage.
As for a red face, this can have a wide variety of causes including rosacea (a skin condition that may also cause swelling and sores), allergies, inflammatory conditions, fever, and sunburn. Medications (including some used to lower blood pressure) can also cause a red face as a side effect. However, one thing that's very unlikely to be behind your red face is high blood pressure. So that's one less thing to get hot and bothered about.