What is the difference between the two? Do they mean the same thing?
If you keep standing there, people will start to look/start looking at you.
Both structures have the same meaning. However, it's preferable to use the infinitive one, when you use the present participle of the verb "start", e.g.:
Looking at your sentence, both can be used as you're not using the present participle form of the verb "start":
If you keep (on) standing there, people will start to look/start looking at you.
I found a web that gives other difference, but that difference doesn't seem to exist actually as far as I know, and @FumbleFingersReinstateMonica indicates below.
I am not a native speaker but as far as I know, they both mean the same thing. There might be a very lil difference that can perhaps only be spotted by native speakers, but grammatically here they are both working as a noun.
Please take my answer as just a sharing of point of view and I am still learning.
I have just found something which I want to share and try to explain. Let's take two sentences I like to go my friend's house and I like going my friend's house. In first sentence, the subject 'I' like 'to go' means I am not going currently but I like or I want to go and the infinite 'to go' is yet to happen in future. I like that future event.
In the 2nd sentence, I like going his house, here, the subject likes 'going' , 'going' is gerund here but still giving continuous form, I like that feeling of going.
But in the case of start, the meaning of the word 'start' manages to fill the difference between two.
Like they will start to look at you...'to look' is yet to happen in future but adding 'will start' before it will begin that future thing.
And, they will start looking at you, 'looking' is continuous form but adding 'will start ' before it begins that continuous form.
So, in this case the difference between them is almost nothing.
And,That' why there are certain verbs which can follow both gerunds and infinitives without changing the meaning of sentences.
I can be wrong too.....so just take it as a sharing of point of view.