Transgender is the current umbrella term for anyone whose gender identity differs from their sex assigned at birth. See the full Transgender page.
Transsexual is an older, more clinical term - historically tied specifically to medical transition - that's now considered largely outdated. It was coined by Magnus Hirschfeld in 1923; "transgender" became the preferred umbrella term in the 1990s.
Some people, especially from older generations or in medical contexts, still use "transsexual" for themselves, so it isn't a term to police in first-person use. As a term applied to someone else or used generally, "transgender" is preferred - see the full breakdown of outdated terms on the Key Distinctions page.