Is there an advantage to investing in my company's 401k over, say, a Roth IRA or a private investment account in blue chip stocks?

With a Traditional 401(k) contribution, you receive an immediate income tax deduction (through payroll). Your contributions have the opportunity to grow tax-deferred until you withdraw them in retirement when your distributions will be treated as taxable income. In 2015, you can contribute up to $18,000 of earnings to your 401(k), plus an additional $6,000 catch-up if you're age 50+
With a Roth IRA, you receive no income tax deduction now but you get the opportunity for tax-free growth and future tax-free withdrawals in retirement. In the event of emergency, you can always withdraw your original Roth IRA contributions without tax or penalty. Only the earnings (if any) will be taxed and penalized if withdrawn before age 59.5 or 5 years in the Roth (whichever is later).
With a Regular Investment Account, you make investments with after-tax money. You report income (interest, dividends, capital gains distributions) and are taxed on it in the year received. You also report realized capitals gains/losses (and pay taxes) once you've sold the investment.
Generally speaking, if your employer offers a matching 401(k) contribution, be sure to at least contribute whatever's necessary to leverage that match. It's "free money" to you.
Beyond that, it's a question of your current income tax bracket vs. anticipated future bracket in retirement. If you're in a much higher income tax bracket now than you'll likely be in retirement, then receiving the income tax deduction now through traditional 401(k) contribution would be of more value to you. If you're in a lower or equivalent tax bracket now vs. what you anticipate in retirement, then emphasizing Roth contributions would typically be of more value to you. 
​I hope that helps. All the best!