Difference-in-differences vs controlling for baseline imbalance on outcome

Question

Let's say we have data from a small 2-arm pilot trial with baseline imbalance. I wish to compare two approaches to the analysis:

1. Regression of endline (post-treatment) outcome data on an indicator of study arm that controls for baseline values on the outcome of interest

2. Difference-in-differences

Here's some toy data that mimics a real example.

# function for simulating data with fixed parameters
# https://stackoverflow.com/a/19343398/841405
  mysamp <- function(n, m, s, lwr, upr, nnorm) {
    samp <- rnorm(nnorm, m, s)
    samp <- samp[samp >= lwr & samp <= upr]
    if (length(samp) >= n) {
      return(sample(samp, n))
    }  
    stop(simpleError("Not enough values to sample from. Try increasing nnorm."))
  }
# load packages
library(tidyverse)

# simulate baseline and endline data based on real-world example
# some loss-to-followup at endline
  set.seed(42)
  base.t <- mysamp(n=32, m=3.03, s=0.46, lwr=0, upr=24, nnorm=1000)
  base.c <- mysamp(n=32, m=4.53, s=0.52, lwr=0, upr=24, nnorm=1000)
  end.t <- mysamp(n=23, m=2.21, s=0.63, lwr=0, upr=24, nnorm=1000)
  end.c <- mysamp(n=22, m=2.23, s=0.39, lwr=0, upr=24, nnorm=1000)

# create long data
  dat <- data.frame(id=c(seq(1:32),           # control, baseline
                         seq(1:22),           # control, 3 month
                         seq(from=33, to=64), # treatment, baseline
                         seq(from=33, to=55)),# treatment, 3 month
                    trt=c(rep(0, 32+22),      # control
                          rep(1, 32+23)),     # treatment
                    end3mo=c(rep(0, 32),      # control, baseline
                             rep(1, 22),      # control, 3 month
                             rep(0, 32),      # treatment, baseline
                             rep(1, 23)),     # treatment, 3 month
                    score=c(base.c,           # control, baseline
                            end.c,            # control, 3 month
                            base.t,           # treatment, baseline
                            end.t))           # treatment, 3 month
# reshape wide
  datw <- 
    dat %>%
    mutate(end3mo = case_when(end3mo==1 ~ "end3mo",
                              TRUE ~ "baseline")) %>%
    group_by(end3mo) %>%
    spread(end3mo, score)

Here's the result for the first approach, regressing endline outcome scores on an indicator of study assignment and controlling for baseline data.

# controlling for baseline 
  summary(lm(end3mo ~ trt + baseline, data=datw))

#Coefficients:
#            Estimate Std. Error t value Pr(>|t|)   
#(Intercept)  2.69332    0.87332   3.084   0.0036 **
#trt         -0.19323    0.31856  -0.607   0.5474   
#baseline    -0.09929    0.19330  -0.514   0.6102   
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Here's the result of the second approach, difference-in-differences:

# difference in differences estimate (interaction)
  summary(lm(score ~ trt*end3mo, data=dat))

#Coefficients:
#            Estimate Std. Error t value Pr(>|t|)    
#(Intercept)  4.55646    0.09029  50.467  < 2e-16 ***
#trt         -1.46102    0.12768 -11.443  < 2e-16 ***
#end3mo      -2.30747    0.14145 -16.313  < 2e-16 ***
#trt:end3mo   1.40694    0.19875   7.079  1.7e-10 ***
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Results:

1. Coefficient on trt is is -0.19323 (Not shown: LOCF for missing data at endline gives trt of 0.39223)
2. DID estimate (trt:end3mo) is 1.4