Can I sue my own lawyer for malpractice in Ontario, and how is that different from appealing my case?

Yes. A lawyer who handles your matter negligently - missing a limitation period, giving clearly wrong advice, botching a closing - can be sued for professional negligence like any other professional, using the same duty, breach, causation, and damages framework. This is a separate track from an appeal. An appeal challenges whether a court or tribunal made a legal error in its decision; a malpractice claim challenges whether your own lawyer's conduct fell below a reasonably competent standard, regardless of what any court decided.

Malpractice claims involving litigation often require proving a "case within a case" - showing that, had your lawyer acted competently, your underlying claim or defence would probably have succeeded, and by how much. That makes these claims evidence-heavy and often expensive to run. You can also file a complaint with the Law Society of Ontario, which regulates lawyer conduct and can discipline a lawyer, but a Law Society complaint does not get you compensation - only a civil claim can do that. Most Ontario lawyers carry mandatory professional liability insurance, which is usually who ends up funding a settlement or judgment.