No.

You cannot cross the border with just a birth certificate and photo ID, as @Evan said. There is one exception, and that is the case of minors (15 or under or 18 and under if with a group). From the US Customs and Border Protection:

Children: Canadian citizens 15 years old and younger arriving by land or sea from contiguous territory may present an original or copy of his or her birth certificate or a Canadian Citizenship Card. Canadian Citizens 16 and older must have a valid passport, unless traveling as part of an organized group described below.

• Groups of Children: Canadian citizen children under age 19 arriving by land or sea from contiguous territory and traveling with a school group, religious group, social/cultural organization, or sports team, may also present an original or copy of his or her birth certificate, a Consular Report of Birth Abroad, a Naturalization Certificate, or a Canadian Citizenship Card.

• The group should provide, on organizational letterhead: The name of the group and supervising adult, a list of the children on the trip, the primary home address, phone number, date of birth, place of birth, and name of at least one parent or legal guardian for each child.

• A written and signed statement of the supervising adult certifying that he or she has obtained parental or legal guardian consent for each participating child.

Sorry, but no. A birth certificate and photo ID are no longer accepted. This page lists the documents that will get you back into the US from Canada.

To answer the second part of your question, to enter Canada you can use any of the following documents (I'm assuming you're a US citizen, emphasis mine)

Visitors to Canada
Identification requirements for United States citizens and permanent residents

If you are a U.S. citizen, ensure you carry proof of citizenship such as a passport, birth certificate, a certificate of citizenship or naturalization, or a Certificate of Indian Status along with photo identification. [...]

Whether you're entering by air, land or water, we recommend you carry a valid passport for all travel abroad, including visits to Canada from the U.S. A passport may be required by your airline or alternative transportation authority, as it is the only universally-accepted identification document.

Citizens of the U.S. who are members of the NEXUS program may present their membership card to the CBSA as proof of identification and as a document that denotes citizenship, when arriving by air, land, or marine modes.