Expressing "strict" with 严肃 and 严格
严肃 and 严格 have the same translation but they are used differently.
The adjective 严肃
The adjective 严肃 
emphasizes that a person performs his duties or acts conscientiously.
emphasizes that a person performs his duties or acts conscientiously. Examples
警察严肃跟进违规行为
The police monitors criminal activities seriously
jǐngchá yánsù gēnjìn wéiguī xíngwéi
这个案子要严肃处理
You should be serious about this business
zhè ge ànzi yào yánsù chǔlǐ
严肃的政治家不用这样的方式讲话
Serious politicians do not speak like that
yánsù de zhèngzhìjiā bú yòng zhèyàng de fāngshì jiǎnghuà
严肃 is also expressed with a tone and facial expression which make people feel respect and fear at the same time.
Examples
她的脸显得严肃起来
She became serious
tā de liǎn xiǎn de yánsù qǐlái
他说话的语气变得严肃起来
The tone of his voice became serious
tā shuōhuà de yǔqì biàn de yánsù qǐlái
The adjective 严格
严格 expresses strictness in following the rules. It is often used to talk about discipline, upbringing and management.
expresses strictness in following the rules. It is often used to talk about discipline, upbringing and management.Examples
她妈妈，平时对孩子教育很严格
Her mother usually takes the upbringing of her children seriously
tā māma, píngshí duì háizi jiàoyù hěn yángé
新兵受严格的纪律约束
Recruits follow strict discipline
xīnbīng shòu yángé de jìlǜ yuēshù
这个行业有严格的管理
This industry is strictly regulated
zhè ge hángyè yǒu yángé de guǎnlǐ