When used with another verb 开始 and 起来 both convey the meaning: to start doing something. Are they synonymous in that specific usage?
[开始] (start; begin) itself is a [verb], which means it can be the only verb in a sentence
你可以开始 (you can begin)
会议可以开始 (the meeting can start)
It can work with another co-verb, which means it can co-exist with another verb in a sentence
你可以开始写 (you can start writing)
你可以开始读 (you can start reading)
It is always [开始 + v] never [v + 开始]
[起来] (start) is a [verb particle] that indicates the verb has begun
跑 = run
打 = fight
哭 = cry
跑起来 = start running
打起来 = start fighting
哭起来 = start crying
It is always [v + 起来], never [起来 + v]
The object must be placed after 开始 and its co-verb
- 你可以开始写你的辞职信了 (You can start writing your resignation letter)
[V + 起来] usually don't take objects.
When 起来 is working with a separable compound verb, it can break up the two characters, inserts 起 between them and 来 at the end
他突然唱歌 (he suddenly sings)
他突然唱起歌来 (he suddenly start singing)
他突然说话 (he suddenly speaks)
他突然说起话来 (he suddenly start speaking)
When 起来 is working with an inseparable compound verb, it leaves the compound verb intact
打斗起来 (start fighting) never 打起斗来
工作起来 (start working) never 工起作来
The above two examples can be seen commonly in [topic + comment] sentences
[topic: 两帮人打斗起来 (If the two groups start fighting)] + [comment: 肯定会有死伤 (there will be casualties)]
[topic: 他工作起来(when he start working)] + [comment: 连饭也会忘了吃 (he can even forget to eat]
unrelated to the role of verb particle, 起, and 来 can be used as result complement for the same verb
站 = stand
站起 = stand up
站起来 = standing up (not start standing)
记 = remember
记起 = remember (successfuly)
记起了 = have remembered
I am only a semi-advanced learner so I will depend on others to correct me.
First 开始 and 起来 behave quite differently syntactically and connect to the verb differently as others have pointed out. Also, since 开始 is two syllables, it won't work well with another verb that is only one syllable. I don't think you would say 他开始跑, but rather 他开始跑步 to say that someone has started to run as exercise or 他跑起来 to say "he broke out in a run."
Second, I think that 起来 gives a vivid picture of a sudden transition into a new action; whereas 开始 lacks this meaning and is more appropriate for something gradual or less vivid. For example, I think that if someone started taking singing lessons, you would say 他开始每天唱歌 to indicate they have started a daily practice routine. If you say 他每天唱起歌来, your are saying something like "he breaks out into song every day."
Furthermore, I think 起来 is appropriate only when the new action actually gets perceptibly under way. With 开始, the new action may not be apparent yet.
Think of the line in 义勇军进行曲: 起来！起来！起来！
I certainly would not translate this 起来 as Get up! How lacking in culture!
Try: Arise! Arise! Arise! (definitely not Start! Start! Start!)
You can say: 看起来， but not 看开始
You can say: 开始看， and you can say 起来看
At 6am my cat starts calling me to get up.