A great question! Getting children on the path to saving and understanding money is an important stage of parenting, and opening up that first account is the initial step in that direction. For my own daughter, my wife and I opened two separate accounts:
An interest-bearing savings account at a bank for cash savings, and
A self-directed education savings plan for longer-term investing for education.
(Specifically, we have an RESP, since we are in Canada. Americans should consider a 529 plan.)
The Savings Account: Why?
A savings account at a bank (or credit union) is relatively easy to open. Since it's easy to open, you are much less likely to procrastinate and sit on those cash or cheque gifts that ought to be deposited– or worse, . Worse if they're spent by you instead :-/
It's easy to deposit money into the savings account. Our bank provided an ATM card to use for deposits. Cash or cheque gifts my daughter receives for her birthday, etc. can be deposited directly into her account. You could also set up direct deposit of the Universal Child Care Benefit you may be getting from the Canadian government.
We don't want all of the money locked up in longer-term education savings. There may be things our daughter will need or want that won't qualify as an education savings plan withdrawal.
The Education Savings Plan: Why?
Your child's post-secondary education is going to cost you a lot of money.
The savings account can only hold cash, and interest rates are pitifully low. Whereas the education plan allows us to hold mutual funds, bonds, ETFs, stocks. At the moment we have the education plan diversified across a handful of low-fee index mutual funds. We rebalance every year. However, managing the education plan involves more paperwork than a savings account – but it's worth it.
Many government-sponsored/regulated education savings plan provide tax benefits such as tax sheltered growth and taxation of the eventual income in the hands of the beneficiary.
Specifically for Canada, education savings are subsidized by the Canadian government with a grant (free money!): a 20% match on the first $2500 each year, for children under age 18. , to a maximum of $7200. It adds up!
Withdrawals from these plans can be complicated. That's good, since for education savings it's better to be relatively hands off. If not, the temptation is too great to make a withdrawal for some other purpose, e.g. Disney World :-)
Finally, when it comes to taxation – savings account interest is typically taxable (and in Canada at least, "attribution rules" make interest income taxable in the hands of the parents) – but there's usually so little interest lately that it's a rounding error on your income tax return.
Whereas, the education savings are tax-advantaged, as mentioned above.