Sharjah is perceived as a very conservative place and certainly doesn't have the night-life or the high quality of life that Dubai has, but could also be a quieter and more cost effective option. Many people are opting to live in Sharjah and commute to Dubai every now and then for social obligations, to go shopping, visit malls, etc.
You must also see what provision your company will make for accommodation and transport - the costs are much higher in Dubai if you have to pay for it. Also, there is little available accommodation in the Deira side of Dubai (closest to Sharjah) so you are likely to have difficulties finding something unless your company already has some available apartments/houses.
Finally, you must see how long you intend to stay and what provision you want to make for your child's schooling, etc.
I have tried to sum up the differences below.
Sharjah / Dubai (in blue)
People tend to be a little more reserved in terms of clothing (skimpy tops and shorts, etc.).
Anything goes pretty much. There used to be a law which prevented women showing their waist or arms/ legs but that law has been lost somewhere in the dust!
Alcohol not permitted at all (but nobody cares what you do in your home - unless you are selling it or you are involved in a car accident and it is found in your car).
Alcohol served in 4 & 5 star hotels and some quality clubs. Also, sold in specific stores but only if you have a valid alcohol licence.
Zero tolerance of Driving under the influence of alcohol or other substances in all Emirates.
Night-clubs do not exist.
Night-clubs of every variety exist.
Coffee shops and restaurants of every variety exist but average to good standard - few are high quality.
Every imaginable type of coffee shop and restaurant exist upto and include a huge variety of high quality places.
Women in main shopping areas, beaches, etc. are protected from unwanted harassment by a Social Police but only for cases where youths or other pests harass women and families.
There is some security at malls and the police can be called upon in case of harassment but no direct control.
Most usual stores available and small to medium sized malls available, as well as food courts, cinemas, etc. but can get extremely crowded on weekends esp. with lower income groups.
Almost every imaginable store available but can also get crowded on weekends.
Traffic within the shopping areas can be quite bad most of the day and night.
Traffic fluctuates and can be particularly easy to access shopping areas during the day when most are at work.
Commuting to/from Dubai at peak hours is hell - off peak hours can be a lot easier.
Vast majority of Sharjah residents work in Dubai so the traffic is in the related direction. Commuting is tough at most hours. Main roads have been converted into Highways so it is essential to know the roads well or liable to spend a long time to get to somewhere relatively near.
Working and Living in Sharjah - less commute to and from work.
Working in Sharjah but living in Dubai - commuting against the major traffic flow so can be good.
Amenities and Services quite slow and painful - fewer public officers speak English and are helpful.
Amenities and Services quite developed - rare to find someone who does not speak English.
Cost of Living is lower, including food/clothing/etc.
Cost of Living is higher.
Very few Greeks live in Sharjah, although quite a few Cypriots do.
Most Greeks/Cypriots live in Dubai.
Basic knowledge of Arabic/Hindi can be very helpful.
English is more than sufficient to communicate.
There are a few very good clinics and one good private hospital.
There are many good clinics and a few good private hospitals.
Both Dubai and Sharjah:
Shopping centres usually open upto 10:00 pm on weeknights and may be upto midnight on Thursday/Friday. Shops close around 9:00/10:00 but Supermarkets can be open upto midnight on weekends and some even 24 hours a day every day of the week.
Very friendly to foreigners.
Some medication restricted - eg. Panadol with Codeine (Solpadeine) is banned due to Codeine being considered an addictive substance, even if you have a prescription, best not to bring it in with you - if caught you face deportation and black-listing!