"Please help me buy some things."
it is as you want to specify that there are a few things (due to "some"). If you omit 一些 the number of things can be inferred to be one, some, or many.
请帮助我买一些东西 vs. 请帮我买一些东西 What's the difference? Is the first one incorrect?
Both 帮助 and 帮 are okay. The latter is more concise and maybe just a bit less formal.