Are the polynomial remainder and factor theorems equivalent?

Yes, both are equivalent to the Division algorithm with linear divisor $\,x\!-\!a$. Below, if rings are unfamiliar, let $R[x] = $ polynomials whose coefficients are complex (or real or rational). As for integers, $\, f\bmod g :=\,$ remainder left in $f\div g = \,$ polynomial (long) division with remainder.

TTheorem $\ $ TFAE for a polynomial $\,f\in R[x],\,$ and $\,a\in R\,$ a commutative ring.

$(0)\ \ \ f = (x\!-\!a)q + r\ $ for some $\,q\in R[x],\ r\in R\ \ \ $ [Monic Linear Division Algorithm]

$(1)\ \ \ f\bmod x\!-\!a = f(a)\ \ \ \ \ \ \ $ [Remainder Theorem]

$(2)\ \ \ f(a) = 0\,\Rightarrow\, x\!-\!a\mid f\ \ \ $ [Factor Theorem]

Proof $\ (0\Rightarrow 1)\ \ \ f = (x\!-\!a)q + r\,\overset{\large x\,=\,a}\Longrightarrow\, r=f(a)\,\Rightarrow\,f\bmod x\!-\!a = r = f(a) $

$(1\Rightarrow 2)\ \ \ f\bmod x\!-\!a = f(a) = 0\,\Rightarrow\, x\!-\!a\mid f$

$(2\Rightarrow 0)\ \ \ g := f-f(a)\,$ has $\,g(a) = 0\ $ so $\ g = f-f(a) = (x\!-\!a)q$

Remark $ $ The polynomial division algorithm always works in any polynomial ring for divisors that are monic (lead coef $= 1$ or a unit) such as $\,x-a\,$ above, see here.

The Remainder Theorem implies the factor theorem as noted by John; We can rewrite the statement 'If the polynomial $P(x)$ is divided by $(x-a)$, then the remainder is $P(a)$' as

$$P(x) = (x-a)Q(x)+P(a)$$

where $Q(x)$ is a polynomial of one degree less than $P(x)$. Now, if $P(a)=0$, what do you get?